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On the influence of global cosmological expansion on the dynamics and

kinematics of local systems

Matteo Carrera∗

Institute of Physics, University of Freiburg, Hermann-Herder-Strasse 3, D-79104 Freiburg, Germany

Domenico Giulini†

Max-Planck-Institute for Gravitational Physics, Am Muhlenberg 1, D-14476 Potsdam OT Golm, Germany‡

(Dated: October 15, 2008)

We review attempts to estimate the influence of global cosmological expansion on local systems.Here ‘local’ is taken to mean that the sizes of the considered systems are much smaller thancosmologically relevant scales. For example, such influences can affect orbital motions as wellas configurations of compact objects, like black holes. We also discuss how measurements basedon the exchange of electromagnetic signals of distances, velocities, etc. of moving objects areinfluenced. As an application we compare orders of magnitudes of such effects with the scaleset by the apparently anomalous acceleration of the Pioneer 10 and 11 spacecrafts, which is10−9m/s2. We find no reason to believe that the latter is of cosmological origin. However, thegeneral problem of gaining a qualitative and quantitative understanding of how the cosmologicaldynamics influences local systems remains challenging, with only partial clues being so far providedby exact solutions to the field equations of General Relativity.

PACS numbers:

Contents

I. Introduction 1

II. Strategic outline and results 3A. Improved Newtonian equations 3B. Exact solutions 3

1. Matched solutions 42. Melted solutions 5

C. Kinematical effects 61. Timing and distances 62. Doppler Tracking 6

III. Newtonian approach 7A. Restricted two-body problem in an expanding

universe 7B. Specifying the initial-value problem 8C. Discussion of the reduced effective potential 9

IV. General-relativistic treatment for

electromagnetically-bounded systems 10A. The argument of Dicke and Peebles 10B. Exact condition for non-expanding circular orbits 12

V. General-relativistic treatment for

gravitationally-bounded systems 13A. Spherically-symmetric matchings 13B. The Eisenstaedt theorem 15C. The Einstein–Straus vacuole revisited 15D. The McVittie model 17

1. Interpretation of the McVittie model 182. Motion of a test particle in McVittie spacetime 21

∗Electronic address: matteo.carrera@physik.uni-freiburg.de†Electronic address: domenico.giulini@aei.mpg.de‡Also at:Institute of Physics, University of FreiburgHermann-Herder-Strasse 3, D-79104 Freiburg, Germany

3. Exact condition for non-expanding circular orbitsin McVittie spacetime 22

VI. Kinematical effects 23A. Einstein- versus cosmological simultaneity 23B. Doppler tracking in cosmological spacetimes 25

1. Minkowski spacetime 252. General setting 263. FLRW spacetimes 284. McVittie spacetime 29

VII. Summary and outlook 29

Acknowledgments 30

A. Notation, conventions, and generalities 30

B. Proof of Theorem 1 31

C. Submanifolds 32

D. Spherical symmetry 331. Connection and curvature decomposition 332. Einstein equation in case of spherical symmetry 343. Misner–Sharp energy 354. Spherically symmetric perfect fluids 36

References 37

I. INTRODUCTION

There is by now ample evidence that our Universe isexpanding on average. This means that on the largestscales one observes redshifts from structures that are in-terpreted as recessional motion, also called the Hubbleflow. To first approximation, the relative velocity be-tween two structures grows linearly with their mutualdistance. The constant of proportionality is the so-called

2

Hubble constant, H0, whose value is now fairly accu-rately measured as being close to 70Km · s−1 ·Mpc−1,see e.g. (Komatsu et al., 2008). This means that for anyadditional mega-parsec (Mpc = 3.262× 106 lightyears =3.086×1019 km) the recessional velocity picks up an extra70 kilometers per second. Clearly, typical peculiar veloci-ties superimpose on the global Hubble flow. For galaxiesthey can be up to 1000 kilometers per second, so thatthe Hubble flow definitely dominates at distances above200Mpc, i.e. above supercluster scale. In this respect it isremarkable that Hubble’s classic paper (Hubble, 1929) of1929 plots the velocity-distance relation of extra-galacticnebulae only up to 2Mpc, though it has to be added thatin those days distances where generally underestimated,sometimes up to a factor of 10.

For pedagogical purposes the global expansion is some-times represented by the two-dimensional balloon model,in which three-dimensional space corresponds to the two-dimensional surface of an inflating rubber balloon; seee.g. § 27.5 in (Misner et al., 1973). At each point at-tached to the rubber material an observer sees otherpoints attached in a state of radial recessional motion,the faster the further they are away. This picture is usedto stress that each point is locally (i.e. with respect tothe local rubber material) at rest but receding from allother points because space inbetween is itself expand-ing. However, this global expansion does not affect allstructures: Local overdensities in the matter distribu-tion may inhibit space from expanding. In the balloonmodel of (Misner et al., 1973) this is represented by littlepennies being glued onto the balloon. The rubber ma-terial underneath the coins does not expand due to thestiff glue which holds it in place. The question ariseswhat, in reality, are the structures corresponding to thecoin and what dynamical mechanism provides the glue?It is often heard that ‘bound systems’ do not partici-pate in the global expansion, or that systems below thescale of galaxy clusters ‘break away’ from the Hubbleflow. But what does ‘bound’ and ‘break away’ reallymean?1 For example, is it obvious that the Astronom-ical Unit is not affected by global expansion (compare(Krasinsky and Brumberg, 2004; Standish, 2004)) or canit even be, as e.g. suggested in (Fahr and Siewert, 2008),that our Universe is contracting on small scales while itexpands in the large? If so, what precisely would rule therelation between contracting and expanding scales?

The purpose of this paper is to review and discuss at-tempts that aim to make precise and answer some of thesefundamental questions, taking due account of the dynam-ical laws and the kinematical framework of General Rela-tivity. We will emphasize the changes in kinematical rela-tions within time-dependent spacetime geometries, whichseem to be widely neglected in related discussions.

1 For a recent discussion on the meaning of ‘joining the Hubbleflow’ see (Barnes et al., 2006).

Next to being a question of fundamental inter-est, the raised issue also needs to be clarifiedquantitatively in connection with more practicalaims, like, e.g., the modeling of celestial referenceframes (Klioner and Soffel, 2005). The specific questionof whether the global expansion has any influenceon the local dynamics and kinematics within theSolar System has recently also attracted increasingattention in connection with the so-called ‘Pioneer-Anomaly’ (Anderson et al., 1998; Anderson et al.,2002; Markwardt, 2002; Nieto and Turyshev, 2004;Turyshev et al., 2005a,b), henceforth abbreviated byPA. Here frequency-measurements in Doppler trackingare translated into standard kinematical quantities,like velocity and acceleration. The result shows ananomalous acceleration of the Pioneer satellites directedtowards the center of the Solar System. In (Markwardt,2002) the magnitude of this acceleration is reportedto be a = 8.6 ± 1.34 × 10−10m · s−2. Note that suchan apparently small acceleration amounts to variationsin spatial localization of nearly 500 kilometers after10 years. It so happens that the magnitude of thisacceleration is very close to the product of the currentvalue of the Hubble constant, H0, and the velocity oflight in vacuum:

H0c ≈(70 km · s−1 ·Mpc−1

)(3× 105 km · s−1

)

= 7× 10−10m · s−2 .(1)

Whether this ‘almost coincidence’ of numbers does in-deed have any deeper significance can and should onlybe decided on the basis of reliable estimates within thedynamical framework of General Relativity. There al-ready exist various speculations and claims in the litera-ture that try to attribute the PA to either simple kine-matical (e.g. (Rosales and Sanchez-Gomez, 1998)) or dy-namical (e.g. (Fahr and Siewert, 2008)) effects of a timevarying background geometry, though non of them doesjustice to the requirements posed by General Relativity.2

This is clearly a very difficult task: There is very littleanalytical knowledge of how to model in terms of ex-act solutions, or controlled approximations to such, thehierarchy of mutually embedded systems: Solar System→ Galaxy → Local Group → Cluster → Supercluster→ Standard-Cosmological Solution. Usually we expecteach such system to define a typical length scale beyondwhich we may consider it as quasi isolated (Cox, 2007).But, clearly, whether this is a valid assumption or notcan only be decided on the basis of a self-consistent dy-namical consideration. In our context all this suggests tofirst study the influence of cosmic expansion on the mostsimple systems immersed in an otherwise homogeneouscosmological background. We will see that this already

2 The reader will soon find out that we disagree with all suchclaims.

3

poses a number of non-trivial analytical as well as con-ceptual problems.

In this article we will derive upper bounds for variouseffects of global expansion on local systems in the contextof such simple models. The idea here is that the upperbounds so derived will a fortiori be upper bounds in morerealistic models, since a further embedding of the systemwe consider into a higher structure of local overdensitieswill further suppress the influence of cosmological expan-sion, whereas we assume it to be fully active outside thesystem of interest. Hence upper bounds we derive willremain upper bounds in more realistic situations. In par-ticular, if we find the relevant upper bounds to be outsidecurrent experimental reach, this will maintain to be thecase in more realistic contexts.

II. STRATEGIC OUTLINE AND RESULTS

A. Improved Newtonian equations

The strategies that so far have been followed aretwofold: Either one studies modified Newtonian or spe-cial relativistic equations of motions for two point-particles with a force of mutual attraction (gravita-tional or electromagnetic). The modifications are de-rived from putting the system into a fixed standard-cosmological background (usually spatially flat) withoutback-reactions being taken into account. We shall discussthis approach in Sections III and IV. Our discussion,based on (Carrera and Giulini, 2005), complements theperturbative analysis in (Cooperstock et al., 1998) whichmisses all orbits which are unstable under cosmologicalexpansion (which do exist). In this respect we follow avery similar strategy as, e.g., in the more recent papersby (Price, 2005) (the basic idea of which goes back atleast to (Pachner, 1963, 1964)) and also (Adkins et al.,2007), though we think that there are also useful differ-ences. We also supply quantitative estimates and clarifythat the improved Newtonian equations of motion arewritten in terms of the right coordinates (non-rotatingand metrically normalized). The purpose of this modelis to develop a good physical intuition for the qualitativeas well as quantitative features of any dynamical effectsinvolved.Eventually the Newtonian model just mentioned has

to be understood as a limiting case of a genuinely rel-ativistic treatment. For the gravitational case this isdone in Section V (an alternative and more geometricderivation is given in Section VI.B), where we employthe McVittie metric to model a spherically symmetricmass embedded in a spatially flat Friedmann–Lemaıtre–Robertson–Walker (FLRW) universe. The geodesic equa-tion is then, in a suitable limit, shown to lead to theimproved Newtonian model discussed above (see also(Carrera and Giulini, 2005)). The same holds for theelectromagnetic case, as we show in Section IV. Therewe take a slight detour to also reconsider a classic ar-

gument by Dicke & Peebles (Dicke and Peebles, 1964),which allegedly shows the absence of any relevant dy-namical effect of global expansion. Its original form onlyinvolved the dynamical action principle together withsome simple scaling argument. Since this reference isone of the most frequently cited in this field, and sincethe simplicity of the argument (which hardly involvesany real analysis) is definitely deceptive, we give an in-dependent treatment that makes no use of any hypo-thetical scaling rules for physical quantities other thanspatial lengths and times. Our treatment, which fol-lows (Carrera and Giulini, 2005), also reveals that theoriginal argument by Dicke & Peebles is insufficient todiscuss leading order effects of cosmological expansion.It is therefore also ineffective in its attempt to contra-dict (Pachner, 1963, 1964).

B. Exact solutions

The other approach consists of finding exact solutionsto Einstein’s field equations for an inhomogeneous situa-tion that, in the most simple case, models a single, quasi-localized, non-rotating, electrically neutral inhomogene-ity within a FLRW universe. Using this inhomogeneoussolution as background one can then study the motionof test particles (following geodesics in the backgroundgeometry) and, in particular, the influence of expansionon this motion.

This approach can be subdivided into two strategies.The first tries to literally construct a new exact solutionout of two known ones, so that the new solution con-tains a connected piece from each of the two old onesas isometric submanifolds. These we refer to as matchedsolutions. This is relaxed in the second, more generalstrategy, where the new solution is merely required tosomehow approximate the relevant part of each of the twoold solutions in some region. These we refer to as meltedsolutions. Needless to say that melted solutions offers amuch greater variety for construction than matched ones.However, it is also true that often not much is knownabout the proper physical interpretation of the former.In this respect the matching solutions usually provide amuch clearer picture.

According to the above requirements, in both cases weshall restrict attention to spherically symmetric space-times which, loosely speaking, approximate a FLRWsolution of standard cosmology for ‘large radii’ and anon-charged, non-rotating compact object characterizedby the exterior Schwarzschild solution for ‘small radii’.(Clearly there must be some characteristic radius interms of which ‘large’ and ‘small’ radii are defined.) Also,one often restricts attention to the spatially flat FLRWmodels for simplicity, which also seems justified in viewof current cosmological data which are compatible withspatial flatness.

4

1. Matched solutions

A first approach to the matching idea was initiatedby Einstein and Straus (Einstein and Straus, 1945, 1946)in 1945 and later worked out in more analytical detailby Schucking (Schucking, 1954). Here the matched so-lution is really such that for radii smaller than a cer-tain matching radius, Rv (henceforth called the vac-uole or Schucking radius), it is exactly given by theSchwarzschild solution (exterior for a black hole, exteriorplus interior for a star) and for radii above this radiusit is exactly given by a FLRW universe for dust matterwithout cosmological constant (this can be generalized,see below). The radius Rv is a function of the centralgravitational mass M and the cosmological mass-density, through the latter of which it also depends on the cos-mological time t. It is determined by

4π

3R3

v · = M . (2)

This formula holds for flat as well as curved FLRW mod-els if ‘radius’ is taken to mean ‘areal radius’, the defini-tion of which is that a two-sphere of areal radius R hasa proper surface area of 4πR2. In flat space the arealradius coincides with the proper radius (the geodesic dis-tance between the center and any point on the sphere), sothat 4π

3 R3v is just the proper volume inside the sphere of

radius Rv (cf. Section V.A). However, in backgrounds ofpositive (negative) curvature this expression is smaller(larger) than the proper volume (the proper volumegrows faster (slower) with areal radius) and hence, forgiven , the left-hand side of (2) is also smaller (larger)than the proper mass of the dust contained within asphere of areal radius Rv.Here we recall that the gravitational mass of a lump of

matter is not just proportional to the amount of matter(baryons) in that region. For example, the kinetic energyas well as the gravitational binding energy also contributeto the gravitational mass. This is expressed in formula(D45) of Appendix D.4, where further explanations willbe provided. As is well known, the mathematical char-acterization of appropriate notions of quasi-local gravita-tional mass that would apply to general spacetimes is anotoriously difficult problem to which various attemptsfor solutions exist; see (Szabados, 2004) for the currentstatus. However, in the spherically-symmetric case, towhich we restrict attention, the so-called Misner–Sharpenergy gives a satisfying and convenient concept of ac-tive gravitational mass. Its definition will be given inSection V.A and more details, including its equality invalue to the Hawking mass, are discussed in the Appen-dices D.3 and D.4.The original construction by Einstein and Straus and

its analytical completion by Schuckingwere quite com-plicated. We will give a much simpler and conceptuallyclearer description in Section V.C, using a suitable refor-mulation of the condition for the matching of solutions.However, it is not hard to gain some intuitive under-standing for the matching construction and the value of

Rv as defined by (2). Let us for the moment restrict tothe spatially flat case and consider the homogeneous andisotropic dust-filled universe at some moment of time t.The dust within a 3-ball of proper radiusRv represents anamount of matter of total mass M as given by (2). Nowcompress this amount of matter in a spherically sym-metric fashion until it becomes a compact star or a blackhole. In Newtonian gravity the gravitational field outsidea spherically symmetric mass distribution only dependson the total mass and not on its radial density distri-bution. This is also true in General Relativity, whichis essentially the content of Birkhoff’s theorem.3 Hencethe above compression preserves equilibrium (albeit anunstable one, see below) for the dust particles just out-side the boundary-sphere of radius Rv. For radii smallerthan Rv we have the Schwarzschild solution (which isthe unique non-trivial spherically symmetric vacuum so-lution according to Birkhoff’s theorem) which thereforematches to the FLRW solution for R ≥ Rv at the bound-ary R = Rv where the matter density is discontinuous.The spatial two-sphere R = Rv is comoving with theHubble flow, meaning that its proper surface area growsin case of expansion. Finally, in case of constant positive(negative) spatial curvature, (2) tells us that the matchedSchwarzschild solution has a smaller (larger) mass thanthe mass that the amount of dust represents within theball of areal radius Rv within the FLRW universe.

The Einstein–Straus model can be generalized in sev-eral ways. Instead of cutting out one ball, one cancut several non-overlapping ones and fill in the in-teriors with Schwarzschild geometries of appropriatemasses. For obvious reasons these are sometimes re-ferred to as ‘Swiss-Cheese models’. These, in turn,can be generalized to the cases of nonvanishing cosmo-logical constant (Balbinot et al., 1988) or nonvanishingpressure (Bona and Stela, 1987). Finally, the Einstein–Straus model can be generalized to spherically symmetricbut inhomogeneous Lemaıtre–Tolman–Bondi (LTB) cos-mological backgrounds (Bonnor, 2000).

Since for the Einstein–Straus model the geometrywithin R ≤ Rv is exactly Schwarzschild spacetime, itis clear that any dynamical system situated in this back-ground geometry (no back reaction) does not notice at allthe cosmic expansion that goes on outside the expandingvacuole R = Rv. Hence global expansion can, in princi-ple, be completely inhibited by local inhomogeneities.

There are, however, several severe problems concern-ing the Einstein–Straus approach. First of all, it cannotprovide a realistic model for the environment of smallstructures in our Universe, ‘small’ meaning below thescales of galaxy clusters or superclusters. To see this,apply (2) to a spatially flat universe whose background

3 An elegant proof of Birkhoff’s theorem will appear as a by-product from our considerations in Appendix D.3.

5

matter density is given by the critical density

crit :=3H0

8πG, (3)

where G is Newton’s constant. Then (2) gives

Rv =(RS R2

H

)1/3 ≈(

M

M⊙

)1/3

400 ly , (4)

where

RS :=2GM

c2≈ M

M⊙

3 km , (5)

RH :=c

H0≈ 4Gpc ≈ 1.3× 1023 km , (6)

are the Schwarzschild radius for the mass M and theHubble radius, respectively. M⊙ = 2 × 1030 kg is thesolar mass.For a single solar mass this gives a vacuole radius of

almost 400 lightyears, which is almost two orders of mag-nitude larger than the average distance of stars in ourGalaxy. Therefore, the Swiss-Cheese model cannot ap-ply at the scale of stars in galaxies. This changes asone goes to larger scales. For example, the Virgo clusteris estimated to have a mass of approximately 1015 solarmasses (Fouque et al., 2001)4, which makes its vacuoleradius 105 times larger than that for a single solar mass,so that it is approximately given by 10Mpc. This is justa little smaller than the average distance of groups andclusters of galaxies within the Virgo supercluster. Hencethe Einstein–Straus approach might well give viable mod-els above cluster scales. Similar conclusion can be drawnfor the vacuole construction in LTB spacetimes (Bonnor,2000): There it is argued that the vacuole might be asbig as the Local Group.The Einstein–Straus solution (as well as its general-

ization for LTB spacetimes given by Bonnor) may alsobe criticized on theoretical grounds. An obvious oneis its dynamical instability: slight perturbations of thematching radius to larger radii will let it increase with-out bound, slight perturbations to smaller radii will letit collapse. This can be proven formally (e.g. (Krasinski,1998), Ch. 3 and (Bonnor, 2000)) but it is also ratherobvious, since Rv is defined by the equal and oppositegravitational pull of the central mass on one side andthe cosmological masses on the other. Both pulls in-crease as one moves towards their side, so that the equi-librium position must correspond to a local maximumof the gravitational potential. Another criticism of theEinstein–Straus solution concerns the severe restrictionsunder which it may be generalized to non spherically-symmetric situations; see e.g. (Mena et al., 2002, 2003,2005; Senovilla and Vera, 1997).

4 Their considerations are based on a LTB model for the cluster.

2. Melted solutions

The above discussion shows that the Einstein–Strausapproach does not give us useful information regardingthe dynamical impact of cosmic expansion on structureswell below the scales of galaxy clusters. For this rea-son other exact solutions are sought. In this respect wewish to remind the reader on the following general as-pect: In physics we are hardly ever in the position tomathematically rigorously model physically realistic sce-narios. Usually we are at best either able to provideapproximate solutions for realistic models or exact so-lutions for approximate models, and in most cases ap-proximations are made on both sides. The art of physicsthen precisely consists in finding the right mixture ineach given case. However, in this process our intuitionusually strongly rests on the existence of at least some‘nearby’ exact solutions. Accordingly, one seeks exact so-lutions in General Relativity that, with some degree ofphysical approximation, model a spherically symmetricbody immersed in an expanding universe. However, itis not as easy as one might think at first to character-ize ‘body’ and ‘immersed’.5 Clearly it is associated withsome inhomogeneity in form of a spatial region with anoverdense matter distribution, as compared to that of theapproximately homogeneous distribution far out. But abody should also be quasi-isolated in order to be dis-tinguishable form a mere local density fluctuation withsmooth transition. Typical exact solutions that mod-els the latter are the LTB solutions, in which matter isrepresented by pressureless dust that freely falls into thelocal over-densitised inhomogeneity. In some sense, theseform the other extreme to the Einstein–Straus solutionsin that they make the transition as smooth and mild asone wishes. Here we shall be interested in models thatsomewhat lie inbetween these extremes.

An attempt to combine an interior Schwarzschild so-lution (representing a star) and a flat FLRW universewas made by Gautreau (Gautreau, 1984). Here thematter model consists of two components, a perfectfluid with pressure and equation of state p = p() out-side the star, and the superposition of this with thestar’s dust-matter inside the star. However, Gautreaualso made the assumption that the matter outside thestar moves on radially infalling geodesics, which isonly consistent if the pressure outside is spatially con-stant. Thus one is is reduced to exact FLRW out-side the star (van den Bergh and Wils, 1984) or the LTBmodel. (Further remarks may be found in (Krasinski,1998), e.g. p. 113 and 165.) Other solutions, model-ing a black hole in a cosmological spacetime, have been

5 In a linear theory, the ‘simultaneous presence’ of two structures,like a local inhomogeneity in an ‘otherwise’ homogeneous back-ground, naturally corresponds to the mathematical operation ofaddition of the corresponding individual solutions. In a non-linear theory, however, no such simple recipe exists.

6

given in the literature. However, these solutions modelobjects which are either rotating (Ramachandra et al.,2003; Vaidya, 1977, 1984), charged (Gao and Zhang,2004), or both (Patel and Trivedi, 1982). Surveys on thesubject of cosmological black holes are (Vishveshwara,2000) and (McClure, 2006). Further interesting so-lutions are given in (Rajesh Nayak et al., 2001) andin (Faraoni and Jacques, 2007; Sultana and Dyer, 2005).The solutions proposed in the latter two works can beseen as generalizations of McVittie’s model (McVittie,1933), which we extensively discuss in Section V.D. Acrucial feature of these solutions is, however, that thestrength of the inhomogeneity6 varies in time, whereasfor the McVittie model it remains constant. Thesesolutions are of interest in their own right (for a de-tailed analysis see (Carrera and Giulini, 2008)), but ourgoal here is to focus on the effects due to cosmologi-cal expansion and not on the effects due to an increas-ing strength of the central inhomogeneity. The solu-tion proposed in the former work (Rajesh Nayak et al.,2001) is the melting of a Schwarzschild spacetime inan Einstein’s static universe. This is a purely staticsolution whose properties and geodesics where studiedin (Ramachandra and Vishveshwara, 2002). For our pur-poses, however, this spacetime is not interesting since itis asymptotically an Einstein universe, and hence not inagreement with the present picture of our Universe atlarge scales.

For these reasons in Section V.D we shall pay spe-cial attention to the McVittie model. This contains adistinguished central object in the sense that the masswithin a sphere centered at the inhomogeneity splits intoa piece that comes from the continuously distributed cos-mological fluid (with pressure) and a constant piece thatdoes not depend on the radius of the enclosing sphere;see our Eq. (82). Moreover, the latter piece is also con-stant in time, meaning that the strength of the centralinhomogeneity remains constant. By the way, McVit-tie’s solutions contain the Schwarzschild–deSitter one asa special case, which was recently used in the litera-ture to estimate the effects of cosmological expansionon local systems (Hackmann and Lammerzahl, 2008c;Kagramanova et al., 2006). In Section V.D.2 we showthat in a suitable weak-field and slow-motion approxima-tion the geodesic equation in McVittie spacetime reducesto the improved Newtonian equations discussed earlier.An alternative and more geometric derivation of the im-proved Newtonian equation for the McVittie case is pre-sented in Section VI.B (see Eq. (151)).

6 In Section V.D.1 we will identify the strength of the inhomogene-ity with the Weyl part of the Misner–Sharp energy.

C. Kinematical effects

1. Timing and distances

Neither the improved Newtonian model nor other gen-eral dynamical arguments make any statement aboutpossible kinematical effects, i.e. effects in connection withmeasurements of spatial distances and time durations in acosmological environment whose geometry changes withtime. This is an important issue if one wants to performthe tracking of a spacecraft, that is a ‘mapping out’ of itstrajectory, which basically means to determine its simul-taneous spatial distance to the observer at given observertimes. But we know from General Relativity that theconcepts of ‘simultaneity’ and ‘spatial distance’ are notuniquely defined. This fact needs to be taken due careof when analytical expressions for trajectories, e.g. solu-tions to the equations of motion in some arbitrarily cho-sen coordinate system, are compared with experimentalfindings. In those situations it is likely that different kine-matical notions of simultaneity and distance are involvedwhich need to be properly transformed into each otherbefore being compared. For example, these transforma-tions can result in additional acceleration terms involvingthe product (1). Accordingly, there were claims in theliterature that these kinematical effects could accountfor the PA; see e.g. (Nieto et al., 2005; Nottale, 2003;Palle, 2005; Ranada, 2005; Rosales and Sanchez-Gomez,1998; Rosales, 2002) and also statements to the contrary(Lammerzahl et al., 2006). In Section VI.A, following(Carrera and Giulini, 2005), we will confirm the exis-tence of kinematical acceleration terms proportional toH0c, but they are suppressed with additional powers ofβ = v/c, which renders them irrelevant as far as the PAis concerned.

2. Doppler Tracking

The discussion in Section VI.B is based on(Carrera and Giulini, 2006). We explain in some de-tail the geometric theory for setting up the kinemati-cal framework in which Doppler tracking should be dis-cussed in oder to properly speak of relative velocities andaccelerations. This is a non-trivial issue which is, in ouropinion, not properly appreciated in the literature on thissubject (related general discussions are (Bini et al., 1995;Bolos, 2007)). Using this setting, we show how to derivean exact Doppler-tracking formula for a flat FLRW uni-verse. This we use to give reliable upper bounds for kine-matical effects caused by cosmic expansion. We also dis-cuss generalizations to McVittie spacetime. Even thoughsuch effects exist, they again turn out to be irrelevant forthe PA.

7

III. NEWTONIAN APPROACH

In order to gain intuition we consider a simple boundedsystem, say an atom or a planetary system, immersedin an expanding cosmos. We ask for the effects of thisexpansion on our local system. Does our system expandwith the cosmos? Does it expand only partially? Or doesit not expand at all?

A. Restricted two-body problem in an expanding universe

We consider the dynamical problem of two bodies at-tracting each other via a force with 1/R2 fall-off. For sim-plicity we may think of one mass as being much smallerthan the other one, though this is really inessential. Onemay think of two galaxies, a star and a planet, a planetand a spacecraft, or a (classical) atom given by an elec-tron orbiting around a proton. The system is placed intoan isotropically expanding ambient universe. We wish toknow the leading order influence of the ambient expan-sion onto the relative two-body dynamics.To leading order, the global expansion is described by

the simple linear Hubble law, R = HR, which states thatthe relative radial velocity of two comoving objects at amutual distance R grows proportional to that distance.More precisely, the term ‘distance’ is here understoodas the geodesic distance in the spacetime hypersurfaceof constant cosmological time t between its two intersec-tion points with the two worldlines of the objects consid-ered. H denotes the Hubble parameter, which generallydepends on t but not on space. It is given in terms of thescale parameter, a(t), via H = a/a.

Taking into account H = (a/a)−H2, the accelerationthat results from the Hubble law is simply given by

R|cosm.acc. = HR+HR =a

aR = −q H2R , (7)

where

q := − aa

a2= − a

aH−2 (8)

is the dimensionless deceleration parameter. To get afeeling for the magnitude, we remark that for the cur-rent best-estimates for the parameters H and q, H0 ≈70Km · s−1 ·Mpc−1 and q0 ≈ −0.6 respectively, we geta/a ≈ 3 × 10−36 s−2, which even at Pluto’s distance of40AU merely amounts to a tiny outward pointing accel-eration of 2× 10−23m · s−1.Now note that, in the sense of General Relativity, a

body that is comoving with the cosmological expansionis moving on an inertial trajectory, i.e. it is force free.On the other hand, according to Newton, a dynamicalforce is, by definition, the cause for deviations from iner-tial motion. In the present context this would mean thatdynamical forces are the causes for deviations from themotions described by (7), which suggests that in New-

ton’s law, m~x = ~F , we should make the replacement

R 7→ R− (a/a)R (9)

in order to apply to the (sufficiently slow) motion of in-teracting point masses in an expanding universe. Notethat this also applies to gravitational interactions in aNewtonian approximation in which gravity is consideredto be a force in the above sense.As we will see, the replacement (9) can be can be jus-

tified rigorously in a variety of contexts, like for gravi-tationally bound systems, using the equation of geodesicdeviation in General Relativity. In doing this one mustmake sure that the Newtonian equations of motion arewritten in appropriate coordinates. They must refer toa (locally) non-rotating frame and directly give the spa-tial geodesic distance. This is achieved by using Ferminormal coordinates along the worldline of a geodesicallymoving observer – in our case e.g. the Sun or the pro-ton –, as emphasized in (Cooperstock et al., 1998). Theequation of geodesic deviation in these coordinates nowgives the variation of the spatial geodesic distance to aneighboring geodesically moving object, e.g. a planet orspacecraft. It reads7

d2xk

dτ2+Rk

0l0xl = 0 . (10)

Here the xk are the spatial non-rotating normal co-ordinates whose values directly refer to the properspatial distance. In these coordinates we furtherhave (Cooperstock et al., 1998)

Rk0l0 = −δkl a/a (11)

on the worldline of the first observer, where the overdotrefers to differentiation with respect to the cosmologicaltime, which reduces to the proper time along the ob-server’s worldline.Neglecting large velocity effects (i.e. terms quadratic

or higher order in v/c) we can now write down the equa-tion of motion for the familiar two-body problem. Afterspecification of a scale function a(t), we get two ODEsfor the variables (R,ϕ), which describe the position8 ofthe orbiting body with respect to the central one:

R =L2

R3− C

R2+

a

aR (12a)

R2ϕ = L . (12b)

These are the (a/a)–improved Newtonian equations ofmotion for the two-body problem, where L representsthe (conserved) angular momentum of the planet (or elec-tron) per unit mass and C the strength of the attractive

7 By construction of the coordinates, the Christoffel symbols Γµαβ

vanish along the worldline of the first observer. Since this world-line is geodesic, Fermi–Walker transportation just reduces to par-allel transportation. This gives a non-rotating reference framethat can be physically realized by gyros taken along the world-line.

8 Recall that ‘position’ refers to Fermi normal coordinates, i.e. Ris the radial geodesic distance to the observer at R = 0.

8

force. In the gravitational case C = GM , where M isthe mass of the central body, and in the electromagneticcase, for the electron-proton system, C = e2/m (Gaus-sian unit), where e and m are the electron’s charge andmass, respectively. In Sections V and IV we will showhow to obtain (12) in appropriate limits from the fullgeneral relativistic treatments.We now wish to study the effect the a term has on the

unperturbed Kepler orbits. We start with the obviousremark that this term results from the acceleration andnot just the expansion of the universe.Next we point out that in the concrete physical cases

of interest, the time dependence of this term is negli-gible to a very good approximation. Indeed, puttingf := a/a, the relative time variation of the coefficient

of R in (7) is f/f . For an exponential scale functiona(t) ∝ exp(λt) (vacuum-energy-dominated universe) thisvanishes, and for a power law a(t) ∝ tλ (for exam-ple matter-, or radiation-dominated universes) this is−2H/λ, and hence of the order of the inverse age of theuniverse. If we consider a planet in the Solar System,the relevant time scale of the problem is the period of itsorbit around the Sun. The relative error in the distur-bance, when treating the factor a/a as constant duringan orbit, is hence smaller than 10−9. For atoms it is muchsmaller, of course. In principle, a time varying a/a causeschanges in the semi-major axis and eccentricity of Keplerorbits (Sereno and Jetzer, 2007). But here we shall ne-glect the time-dependence of (7) and set a/a equal to aconstant A. Because of (8) we have A := −q0H

20 . Then

(12a) can be immediately integrated:

1

2R2 + U(R) = E , (13)

where the effective potential is

U(R) =L2

2R2− C

R− A

2R2 . (14)

We will see below that the three parameters (L,C,A) canbe effectively reduced to two.

B. Specifying the initial-value problem

Solutions of (13) and (12b) are specified by initial con-

ditions (R, R, ϕ, ϕ)(t0) = (R0, V0, ϕ0, ω0) at the initialtime t0. The discussion of the dynamical behavior of Ris most effectively done in terms of the effective poten-tial. Moreover, since perturbations are best discussed interms of dimensionless parameters, we also introduce alength scale and a time scale that appropriately charac-terize the dynamical perturbation and the solution to beperturbed.The length scale is defined as the radius at which the

acceleration due to the cosmological expansion has thesame magnitude as the two-body attraction. This hap-

pens precisely at the critical radius

Rc :=

(C

|A|

)1/3

. (15)

For R < Rc the two-body attraction dominates, whereasfor R > Rc the effect of the cosmological expansion is thedominant one.In order to gain an understanding of the length scales

of the critical radius it is instructive to express it in termsof the physical parameters. In the case of gravitationalinteraction we have C/|A| = GM/(|q0|H2

0 ) and thus

Rc =

(RSR

2H

2|q0|

)1/3

. (16)

Inserting the approximate value q0 = −1/2 of the presentepoch, this reduces to the Schucking radius (4).In the electromagnetic case, e.g. for an electron-proton

system, we have C/|A| = (e2/m)/(|q0|H20 ). Defining, in

analogy with (5), the length scale

Re :=2e2

mc2≈ 5.64 · 10−15m , (17)

the critical radius (15) becomes

Rc =

(ReR

2H

2|q0|

)1/3

≈ 30AU , (18)

where in the last step we inserted q0 = −1/2. This isabout as big as the Neptune orbit!From (16) and (18) one sees that, in both cases, a larger

(smaller) |q0| implies a smaller (larger) critical radius,according to expectations.So much for the length scale. The time scale is defined

to be the period of the unperturbed Kepler orbit (a solu-tion to the above problem for A = 0) of semi-major axisR0. By Kepler’s third law it is given by

TK := 2π

(R3

0

C

)1/2

. (19)

It is convenient to introduce two dimensionless param-eters which essentially encode the initial conditions R0

and ω0.

λ :=

(ω0

2π/TK

)2

=L2

CR0, (20)

α := sign(A)

(R0

Rc

)3

= AR3

0

C. (21)

For close to Keplerian orbits λ is close to one. For reason-ably sized orbits α is close to zero. For example, in theSolar System, where R0 < 100 AU, one has |α| < 10−16.For an atom whose radius is smaller than 104 Bohr-radiiwe have |α| < 10−57.Now, defining

x(t) := R(t)/R0 , (22)

9

equations (13) and (12b) can be written as

1

2x2 + (2π/TK)2 uλ,α(x) = e (23)

x2ϕ = ω0 , (24)

where e := E/r20 now plays the role of the energy-constant and where the reduced two-parameter effectivepotential uλ,α is given by

uλ,α(x) :=λ

2x2− 1

x− α

2x2 . (25)

The initial conditions now read

(x, x, ϕ, ϕ)(t0) = (1, V0/R0, ϕ0, ω0) . (26)

The point of introducing the dimensionless variables isthat the three initial parameters (L,C,A) of the effectivepotential could be reduced to two: λ and α. This will beconvenient in the discussion of the potential.

C. Discussion of the reduced effective potential

1 2 3 4 5 6

-2

-1

0

1

2

3

α = −1 α = −0.3

α = 0

α = 1/4α = 0.6α = 1

FIG. 1 The figure shows the effective potential uλ,α for cir-cular orbits, for which λ = 1 − α for some values of α. Theinitial conditions are x = 1 and x = 0 (see (22)). At x = 1 thepotential has an extremum, which for α < 1/4 is a local mini-mum corresponding to stable circular orbits. For 1/4 ≤ α < 1these become unstable.

Circular orbits correspond to extrema of the effectivepotential (14). Expressed in terms of the dimensionlessvariables this is equivalent to u′

λ,α(1) = −λ+ 1− α = 0.

By its very definition (20), λ is always nonnegative, im-plying α ≤ 1. For negative α (decelerating case) this is

always satisfied. On the contrary, for positive α (acceler-ating case), this implies, in view of (21), the existence ofa critical radius, given by Rc, beyond which no circularorbit exists. These orbits are stable if the considered ex-tremum is a true minimum, i.e. if the second derivativeof the potential evaluated at the critical value is positive.Now, u′′

λ,α(1) = 3λ−2−α = 1−4α, showing stability for

α < 1/4 and instability for α ≥ 1/4. For the acceleratingcase, in view of (21), this implies that the circular orbitsare stable iff R0 is smaller than the critical value

Rsco := (1/4)1/3Rc ≈ 0.63Rc , (27)

where ‘sco’ stands for ‘stable circular orbits’.Summarizing, we have the following situation: in the

decelerating case (i.e. for negative α or, equivalently, fornegative A) stable circular orbits exist for every radiusR0; one just has to increase the angular velocity by someamount stated below in (28). On the contrary, in theaccelerating case (i.e. for positive α, or, equivalently, forpositive A), we have three regions:

• R0 < Rsco, where circular orbits exist and are sta-ble,

• Rsco ≤ R0 ≤ Rc, where circular orbits exist but areunstable, and

• R0 > Rc, where no circular orbits exist.

Generally, there exist no bounded orbits that extend be-yond the critical radius Rc, the reason being simply thatthere is no R > Rc where U

′(R) > 0. Bigger systems willjust be slowly pulled apart by the cosmological acceler-ation and approximately move with the Hubble flow atlater times.9 Modifications of this strict qualitative dis-tinction implied by time dependencies of A in (14) werediscussed in (Faraoni and Jacques, 2007).Turning back to the case of circular orbits, we now

express the condition for an extrema derived above, λ =1− α, in terms of the physical quantities, which leads to

ω0 = (2π/TK)√

1− sign(A)(R0/Rc)3 . (28)

This equation says that, in order to get a circular or-bit, our planet, or electron, must have a smaller or big-ger angular velocity according to the universe expandingin an accelerating or decelerating fashion, respectively.This is just what one would expect, since the effect ofa cosmological ‘pulling apart’ or ‘pushing together’ mustbe compensated by a smaller or larger centrifugal forcesrespectively, as compared to the Keplerian case. Equa-tion (28) represents a modification of the third Keplerlaw due to the cosmological expansion. In principle thisis measurable, but it is an effect of order (R0/Rc)

3 and

9 This genuine non-perturbative behavior was not seen in the per-turbation analysis performed in (Cooperstock et al., 1998).

10

hence very small indeed; e.g. smaller than 10−17 for aplanet in the Solar System.Instead of adjusting the initial angular velocity as

in (28), we can ask how one has to modify r0 in or-der to get a circular orbit with the angular velocityω0 = 2π/TK. This is equivalent to searching the min-imum of the effective potential (25) for λ = 1. This con-dition leads to the fourth order equation αx4 −x+1 = 0with respect to x. Its solutions can be exactly writ-ten down using Ferrari’s formula, though this is not il-luminating. For our purposes it is more convenient tosolve it approximatively, treating α as a small pertur-bation. Inserting the ansatz xmin = c0 + c1α + O(α2)we get c0 = c1 = 1. This is really a minimum sinceu′′1,α(xmin) = 1 + O(α) > 0. Hence we have

Rmin = R0

(

1 + sign(A)

(R0

Rc

)3

+O(

(R0/Rc)6))

(29)This tells us that in the accelerating (decelerating) casethe radii of the circular orbits with ω0 = 2π/TK be-comes bigger (smaller), again according to physical ex-pectation. As an example, the deviation in the radiusfor an hypothetical spacecraft orbiting around the Sunat 100 AU would be just of the order of 1 mm. Since itgrows with the fourth power of the distance, the devia-tion at 1000 AU would be of the order of 10 meters.

IV. GENERAL-RELATIVISTIC TREATMENT FOR

ELECTROMAGNETICALLY-BOUNDED SYSTEMS

In this section we show how to arrive at (12) from arelativistic treatment of an electromagnetically boundedtwo-body system embedded (without back-reaction) intoan expanding (spatially flat) universe. This impliessolving Maxwell’s equations in the cosmological back-ground (30) for an electric point charge (the proton)and then integrate the Lorentz equations for the motionof a particle (electron) in a bound orbit (cf. (Bonnor,1999)). Equation (12) then appears in an appropri-ate slow-motion limit. However, in oder to relate thisstraightforward method to a famous argument of Dicke& Peebles, we shall proceed by taking a slight detourwhich makes use of the conformal properties of Maxwell’sequations.

A. The argument of Dicke and Peebles

In reference (Dicke and Peebles, 1964) Dicke & Pee-bles presented an apparently very general and elegantargument that purports to show the insignificance of anydynamical effect of cosmological expansion on a local sys-tem that is either bound by electromagnetic or gravita-tional forces and which should hold true at any scale.Their argument involves a rescaling of spacetime coor-dinates, (t, ~x) 7→ (λt, λ~x) and certain assumptions on

how other physical quantities, most prominently mass,behave under such scaling transformations. For exam-ple, they assume mass to transform like m 7→ λ−1m.However, their argument is really independent of suchassumptions, as we shall show below. We work from firstprinciples to clearly display all assumptions made.

We consider the motion of a charged point particle inan electromagnetic field. The whole system, i.e. particleplus electromagnetic field, is placed into a cosmologicalFLRW-spacetime with flat (k = 0) spatial geometry. Thespacetime metric reads

g = c2 dt2 − a2(t)(dr2 + r2 gS2) , (30)

where

gS2 = dθ2 + sin2 θ dϕ2 (31)

denotes the metric on the unit two-sphere in standardcoordinates. We introduce conformal time, tc, via

tc = f(t) :=

∫ t

k

dt′

a(t′), (32)

by means of which we can write (30) in a conformally flatform

g = a2c(tc)(c2 dt2c − dr2 − r2 gS2) = a2c(tc)η , (33)

where η denotes the flat Minkowski metric. Here wewrote ac to indicate that we now expressed the expansionparameter a as function of tc rather than t, i.e.

ac := a ◦ f−1 . (34)

The electromagnetic field is characterized by the tensorFµν , comprising electric and magnetic fields:

Fµν =

(0 En/c

−Em/c −εmnjBj

)

. (35)

In terms of the electromagnetic four-vector potential,

Aµ = (ϕ/c,− ~A), one has

Fµν = ∂µAν − ∂νAµ = ∇µAν −∇νAµ , (36)

so that, as usual, ~E = −~∇φ− ~A. The expression for thefour-vector of the Lorentz-force of a particle of chargee moving in the field Fµν is e Fµ

νuν , where uµ is the

particle’s four velocity.

The equations of motion for the system Particle + EM-Field follow from an action which is the sum of the actionof the particle, the action for its interaction with theelectromagnetic field, and the action for the free field, allplaced in the background (30). Hence we write:

S = SP + SI + SF , (37)

11

where

SP = −mc2∫

z

dτ = −mc

∫√

g(z′, z′) dλ , (38a)

SI = − e

∫

z

Aµ dxµ = − e

∫

Aµ(z(λ))z′µ dλ

= −∫

d4xAµ(x)

∫

dλ e δ(4)(x − z(λ)) z′µ ,

(38b)

SF =−1

4

∫

d4x√−g gµαgνβ FµνFαβ

=−1

4

∫

d4x ηµαηνβ FµνFαβ . (38c)

Here λ is an arbitrary parameter along the worldline z :λ 7→ z(λ) of the particle, and z′ the derivative dz/dλ.The differential of the proper time along this worldline is

dτ =√

g(z′, z′) dλ =√

gµν(z(λ))dzµ

dλdzν

dλ dλ . (39)

It is now important to note that 1) the background met-ric g does not enter (38b) and that (38c) is conformallyinvariant (in 4 spacetime dimensions only!). Hence theexpansion factor, a(tc), does not enter these two expres-sions. For this reason we could write (38c) in terms ofthe flat Minkowski metric, though it should be kept inmind that the time coordinate is now given by conformaltime tc. This is not the time read by standard clocksthat move with the cosmological observers, which rathershow the cosmological time t (which is the proper timealong the geodesic flow of the observer field ∂/∂t).The situation is rather different for the action (38a)

of the particle. Its variational derivative with respect toz(λ) is

δSp

δzµ(λ)= −mc

{12gαβ,µ z

′αz′β√

g(z′, z′)− d

dλ

[

gµαz′α

√

g(z′, z′)

]}

.

(40)We now introduce the conformal proper time, τc, via

dτc = (1/c)√

η(z′, z′) dλ = (1/ca)√

g(z′, z′) dλ . (41)

We denote differentiation with respect to τc by an over-dot, so that e.g. z′/

√

g(z′, z′) = z/ca. Using this to

replace z′ by z√

g(z′, z′)/ca and also g by a2η in (40)gives

δSp

δzµ(λ)=

√

g(z′, z′)

acma

{ηµαz

α − Pαµ φ,α

}(42)

where we set

a =: exp(φ/c2) and Pαµ := δαµ − zαzν

c2ηνµ . (43)

Recalling that δSP =∫ δSp

δzµ(λ) δzµdλ =

∫ δSp

δzµ(τc)δzµdτc

and using (41), (42) is equivalent to

δSp

δzµ(τc)= ma

(zα − Pα

µ φ,α

), (44)

where from now on we agree to raise and lower indices us-ing the Minkowski metric, i.e. ηµν = diag(1,−1,−1,−1)in Minkowski inertial coordinates.Writing (38b) in terms of the conformal proper time

and taking the variational derivative with respect to z(τc)leads to δSI/δz

µ(τc) = −eFµαzα, so that

δS

δzµ(τc)= ma

(zµ − Pα

µ φ,α

)− e Fµαz

α . (45)

The variational derivative of the action with respect tothe vector potential A is

δS

δAµ(x)= ∂αF

µα(x)−e

∫

dτc δ(x−z(τc)) zµ(τc) . (46)

Equations (45) and (46) show that the fully dynamicalproblem can be treated as if it were situated in static flatspace. The field equations that follow from (46) are justthe same as in Minkowski space. Hence we can calcu-late the Coulomb field as usual. On the other hand, theequations of motion receive two changes from the cos-mological expansion term: the first is that the mass mis now multiplied with the (time-dependent!) scale fac-tor a, the second is an additional scalar force induced bya. Note that all spacetime dependent functions on theright hand side are to be evaluated at the particle’s loca-tion z(τc), whose fourth component corresponds to ctc.Hence, writing out all arguments and taking into accountthat the time coordinate is tc, we have for the equationof motion

zµ =e

mac(z0/c)Fµ

α(z)zα

−(−c2ηµα + zµzα

)∂α ln ac(z

0/c) (47a)

=e

mac(z0/c)Fµ

α(z)zα

−(−cηµ0 + zµz0/c

)a′c(z

0/c)/ac(z0/c) , (47b)

where a′c is the derivative of ac.So far no approximations were made. Now we write

zµ = γ(c, ~v), where ~v is the derivative of ~z with respectto the conformal time tc, henceforth denoted by a prime,and γ = 1/

√

1− v2/c2. Then we specialize to slow mo-tions, i.e. neglect effects of quadratic or higher powers inv/c (special relativistic effects). For the spatial part of(47b) we get

~z ′′ + ~z ′ (a′c/ac) =e

mac

(~E + ~z ′ × ~B

), (48)

where we once more recall that the spatial coordinatesused here are the comoving (i.e. conformal) ones and theelectric and magnetic fields are evaluated at the particle’sposition ~z(tc).From the above equation we see that the effect of cos-

mological expansion in the conformal coordinates showsup in two ways: first in a time dependence of the masswhich scales with ac, and, second, in the presence a fric-tion term. Dicke & Peebles neglect the friction term

12

and simply conclude as follows: In the adiabatic approx-imation, which is justified if typical time scales of theproblem at hand are short compared to the world-age(corresponding to small ε2 in (98b)), the time-dependentmass term leads to a time varying radius in comoving(or conformal) coordinates of r(tc) ∝ 1/ac(tc). Hencethe physical radius (given by the cosmological geodesi-cally spatial distance), r∗ = acr, stays constant in thisapproximation. Hence, within this approximation, elec-tromagnetically bound systems do not feel any effect ofcosmological expansion.But what does ‘this approximation’ refer to? We will

see that it amounts to neglecting precisely the leadingorder contributions. This is easy to see if we cast (48)into physical coordinates, given by the cosmological timet and the cosmological geodesic spatial distance r∗ :=a(t)r. We have dtc/dt = 1/a and the spatial geodesiccoordinates are ~y := a(t)~z. Denoting by an overdot thetime derivative with respect to t, the two terms on theleft hand side of (48) become

~z ′′ = a(~yH2 − ~yH) + a(~y − ~y a/a) , (49a)

~z ′(a′c/ac) = −a(~yH2 − ~yH) , (49b)

where H = a/a. This shows that the friction term can-cels against the first-order derivative terms in ~y and athat one gets in rewriting the left-hand side of (48) inphysical coordinates.10 The only additional term nextto ~y that survives is precisely the familiar accelerationterm (7). Inserting (49) into (48), whose right-handside we now specialize to a pure electric Coulomb field,~E(~z) = Q~z/|~z |3 and ~B(~z) = 0, we arrive at

~y − ~y (a/a) =eQ

m|~y|3 ~y . (50)

After introducing polar coordinates in the orbital planewe exactly get (12).

B. Exact condition for non-expanding circular orbits

In (Bonnor, 1999) a necessary and sufficient condi-tion for the existence of non-expanding orbits is derivedfor the electron-proton system in a spatially flat FLRWspacetime. Here ‘non-expanding’ is defined as of con-stant areal radius. This condition follows directly fromthe Lorentz equation of motion for the electron in theexternal electric field of the proton, the normalizationcondition of the electron’s four-velocity, and the condi-tion of constancy of the areal radius. In our notation, in-troducing the dimensionless quantities h(t) := RH(t)/c,

10 Since the friction term cancels, the critical remark [27] in(Adkins et al., 2007) regarding its magnitude is based on a mis-understanding.

l := L/Rc, and µ := Re/2R, the conditions for the exis-tence of non-expanding circular orbits reads as follows:

R

ch =

(1− h2)3/2

(1 + l2)1/2

(

µ− l2 + h2

√

(1 + l2)(1 − h2)

)

. (51)

Recall that Re is defined in (17) and H(t) and L denotethe Hubble function and, respectively, the (conserved)electron’s angular momentum per unit mass. The abovecondition is a first-order autonomous ODE for the func-tion h(t), and hence for the Hubble function H(t). Thisis the constraint on the spacetime (more precisely, onthe scale factor a(t)) that one gets by imposing the ex-istence of non-expanding circular orbits for two oppo-sitely charged point masses. If such orbits exist, (51)amounts to the generalization of Kepler’s third law toFLRW spacetimes, which here gives a relation betweenthe scale function on one hand and the orbital parame-ters R and L as well as the field-strength parameter Re

on the other. Recall that in Newtonian physics the thirdKepler law is, in our notation, simply given by l2 = µ.The easiest solutions of (51) are of course the station-

ary ones, that is with h(t) ≡ h0, for some constant h0.This means that the scale factors is exponentially ex-panding,

a(t) = a0 exp(H0t) , (52)

where H0 := h0c/R and a0 is some positive constant. Inother words, the spacetime is given by the de Sitter so-lution (Λ-dominated universe). In this case (51) reducesto

l2 + h20

√

(1 + l2)(1− h20)

= µ . (53)

Notice that a larger Hubble parameter, hence a largerh0, makes the l.h.s. larger. Consequently, (53) tells usthat with a larger Hubble parameter we must give to theelectron a smaller angular velocity (smaller l) in order tokeep it on a non-expanding circular orbit with the sameradius. This, according to intuition, is in order to com-pensate the extra cosmological pull with a reduced cen-trifugal term. In case of Minkowski spacetime (h0 = 0)

the above relation reads l2/√1 + l2 = µ, hence one can

interpret the factor 1/√1 + l2 as a special-relativistic cor-

rection to the Newtonian relation l2 = µ. The largest ra-dius at which, in an FLRW spacetime with exponentially-growing scale factor, there is a non-expanding orbit fol-lows from (53) in the limit l → 0. In this limit the condi-

tion reduces to h20/√

1− h20 = µ, which, for small param-

eters h0, simplifies to h20 ≈ µ. Solving for R this gives

the radius (ReR2H/2)1/3, which, taking into account that

q0 = −1 because of (52), exactly corresponds to the crit-ical radius (18).The other (non-stationary) solutions of (51) can also be

found. After separation of variables and an elementaryintegration one gets t as function of h in terms of trigono-metric functions composed with inverse hyperbolic func-

13

tions. This exact expression is again not very illuminat-ing and cannot generally be explicitly inverted so as toobtain h in terms of elementary functions of t. How-ever, if we make use of the smallness of the parametersµ, l2, and h2, a leading-order expansion in these quanti-ties gives a much simpler expression for t(h) which canbe explicitly inverted. In fact, this approximate solutionh(t) is obtained much quicker by solving (51) with theright-hand side being replaced with its leading-order ex-pansion in the mentioned quantities, that is, by solving

R

ch = µ− l2 − h2 . (54)

Here µ− l2 is a constant which depends on the orbit pa-rameters. One must now distinguish between three cases:(a) µ − l2 =: κ2 > 0 for some positive κ, (b) µ − l2 =:−ν2 < 0 for some positive ν, and (c) µ − l2 = 0. Re-calling the Newtonian relation l2 = µ, orbits in the threecases have an angular momentum which is, respectively,smaller, bigger, and equal to the Newtonian one. In-tegrating (54) we get, putting w.l.o.g. t0 = 0, h(t) =κ tanh(κct/R), h(t) = −ν tan(νct/R), and h(t) = R/ct,for the cases (a), (b), and (c), respectively. Then, inte-grating once and exponentiating the result, we get thecorresponding scale functions:

(a) Case µ − l2 =: κ2 > 0 (non-expanding orbits havesub-Newtonian angular momentum)

a(t) = a0 cosh

(κct

R

)

, t ∈ [0,∞) . (55a)

(b) Case µ − l2 =: −ν2 < 0 (non-expanding orbits havesuper-Newtonian angular momentum)

a(t) = a0 cos

(νct

R

)

, t ∈[

0,πR

2νc

)

. (55b)

(c) Case µ − l2 = 0 (non-expanding orbits have Newto-nian angular momentum)

a(t) = a0t , t ∈ (0,∞) . (55c)

In all three cases (a), (b), and (c) a0 is a positive con-stant and the acceleration term a/a is a constant which ispositive, negative, and zero, respectively. Hence, as onewould intuitively expect, the non-expanding orbits havean angular momentum which is smaller, larger, or equalthe Newtonian one, depending on whether the accelera-tion factor a/a is positive, negative, or zero.

V. GENERAL-RELATIVISTIC TREATMENT FOR

GRAVITATIONALLY-BOUNDED SYSTEMS

As advertised in Section II.B, we now wish to dis-cuss exact solutions that may represent quasi-isolated

spherically-symmetric gravitating systems ‘embedded’into cosmological spacetimes. As regards the meaningof ‘embedded’ we distinguish between the strategies of‘matching’ and ‘melting’, as outlined in Sect. II.B.

A. Spherically-symmetric matchings

The complexity an non-linearity of Einstein’s equa-tions make it a very difficult task to construct a suitablevariety of exact solutions which serve as realistic modelsfor actual physical situations. Often exact solutions areonly known for highly idealized situations, typically withhigh degrees of symmetry, in which the field equationssufficiently simplify. One way to construct new solutions(in a suitable sense, see below) from old ones is to gluethem across suitably chosen hypersurfaces along whichthe matter distribution may become singular due to sur-face layers. This approach was pioneered by Lanczos inthe early 1920s (Lanczos, 1924) and put into geometricform by Darmois (Darmois, 1927) and (Israel, 1966); seealso § 21.13 of (Misner et al., 1973). In this section, un-der the assumption of spherically symmetry, we presenta new alternative set of conditions which are equivalentto the old ones. The new conditions only involve scalarquantities, are easy to verify, and have good physical in-terpretations. More details are contained in (Carrera,2008).Here we shall restrict to piecewise continuous matter

distributions without singular (δ-distribution like) sur-face layers, as e.g. in the presence of stars with sharplydefined surfaces. Einstein’s equations can the be satisfiedfor piecewise twice continuously differentiable fields, ifthe field equations at the location of the matching hyper-surface are replaced by their one-dimensional ε-intervalintegrals in normal direction to the hypersurface. Thecondition that two twice continuously differentiable solu-tions (in the ordinary sense) can be matched into a piece-wise twice continuously differentiable solution (in the re-interpreted sense just explained) is then simply given bythe so-called Darmois junction conditions (DJC):

Darmois junction conditions (DJC). For a non-null matching hypersurface Γ , (i) the induced metric gΓand (ii) the extrinsic curvature KΓ shall be continuousthrough Γ .

Let us pause for a moment to say a few more wordsabout the notion of ‘continuity through Γ ’. Gluing to-gether two pieces of spacetimes means the following:Initially one has two spacetimes, say (M+, g+) and(M−, g−), with oriented boundaries Γ+ and Γ−, respec-tively. Given a diffeomorphism φ : φ : Γ+ → Γ− betweenthe boundaries, the glued spacetime is the quotient of thedisjoint union of M+ and M− under the identificationof each point of p ∈ Γ+ with φ(p) ∈ Γ−. The match-ing hypersurface Γ is now the common image of Γ+ andΓ− after identification in the quotient spacetime. Now,a tensor field T is said to be continuous through Γ if

14

T |Γ+ equals T |Γ− under the push-forward action of thediffeomorphism φ, hence if φ∗(T |Γ+) = T |Γ− .Let us now return to the DJC and, in particular, their

physical interpretation. If n is a continuous choice ofunit normal of Γ , it implies that

T (n, ·) is continuous through Γ . (56)

This follows directly from the expressions (C8) of theEinstein tensor given in Appendix C. If Γ is time-like and hence n spacelike, (56) just states the continu-ity of the normal components for the energy-momentumflux-densities, whereas their tangential components to-gether with the energy density may jump across Γ . Inthe absence of surface layers this continuity conditionis just a physically obvious consequence of local energy-momentum conservation, whereas jumps in, say, theenergy-density must clearly be allowed for. For com-pleteness we note that for a spacelike matching surface(56) states the continuity of the densities of energy andmomentum as measured by an observer moving along n(taken to be future pointing), whereas the correspondingcurrents may jump.Let now restrict our attention to spherical symmetric

spacetimes glued along hypersurfaces of spherical symme-try. This means that the latter are left invariant, as set,under the action of SO(3). We recall that the structureof a spherically symmetric spacetime is that of a warpedproduct M = B ×R S2 of a two-dimensional Lorentzianmanifold B and the two-sphere by means of the warp-ing function, R : B → R+, called the areal radius. Thematching hypersurfaces are then of the form Γ = γ×RS

2,where γ is the projection of Γ into B and is called thematching curve. The DJC should then reduce to appro-priate conditions along the curve γ. Indeed, Theorem 1below shows that, in the spherically symmetric case, theDJC are equivalent to the following

Spherically symmetric junction conditions(SSJC). Let Γ be a smooth, non-null, spherically sym-metric matching hypersurface between two sphericallysymmetric spacetimes and n a continuous choice of unitnormal vector field on Γ . Denote with γ the projectionof Γ onto B. Moreover, let v the (unique up to a sign)spherically symmetric, unit vector field on Γ orthogonalto n. The following four functions

(i) the arc-length of γ,(ii) the extrinsic curvature of γ in B: g(n,∇vv),(iii) the areal radius R,(iv) the Misner–Sharp energy E,

shall be continuous through the matching curve γ.

The Misner–Sharp (MS) energy (Misner and Sharp,1964) and (Hernandez and Misner, 1966), which we willdenote by E, is a concept of quasi-local mass that canbe defined in presence of spherical symmetry, and whichproves useful for computational and interpretational pur-poses. It is a function defined in purely geometricalmanner as follows. Given a point p of spacetime, com-pute the sectional curvature of the plane tangent to the

two-dimensional SO(3)-orbit through p and multiply thiswith minus11 one-half of the third power of the areal ra-dius:

E := − 12R

3 Riem |S2S2 . (57)

From (D9) we immediately read off

E =R

2

(1 + 〈dR,dR〉

), (58)

which provides a convenient expression for the computa-tion of the MS energy. We shall show in Appendix D thatthe MS energy is the charge of a conserved current andhow it depends on the energy-momentum tensor for thematter. There we will also briefly discuss its Newtonianlimit. This allows to interpret it as amount of active grav-itational energy contained in the interior of the sphere ofsymmetry (SO(3)-orbit) through p. There we also showthat the MS energy at p is equal to the Hawking quasi-local mass of the two-sphere of symmetry (SO(3)-orbit)through p and hence converges to the Bondi-mass at nullinfinity and, in an asymptotically flat spacetime, to theADM mass at spatial infinity (for the latter two issuessee (Szabados, 2004) and also (Hayward, 1996)). More-over, we will give the decomposition of the MS energy inits Ricci and Weyl parts; see (D34).The name ‘Misner–Sharp energy’ seems now to be

established in the literature, however one should saythat this mass concept goes back at least to (Lemaıtre,1933)12, which gives a coordinate expression forit. Its geometric definition (57) was first givenin (Hernandez and Misner, 1966) and its interpretationas the charge of a conserved current was first derivedin (Kodama, 1980). Later, an alternative definition wasgiven in (Zannias, 1990): There it is showed that the MSenergy can defined in terms of the norm of the Killingfields generating the isometry group SO(3), leading di-rectly to (58). Further relevant studies of the MS en-ergy are (Cahill and McVittie, 1970a,b) and, more re-cently, (Burnett, 1991; Hayward, 1996, 1998).Some comment are needed on the above SSJC. First we

note that, since n and v are spherically symmetric andhence tangent to B we have in view of (D3) and (C6)that g(n,∇vv) = gB(nB,

B∇vB

vB) = ε(nB)KB

γ (vB,vB).Hence, the quantity in (ii) is indeed (up to a possiblesign) the extrinsic curvature of the curve γ in B. Second,note that this quantity, being quadratic in v, does not de-pend on the sign choice of v. Third, since the matchinghypersurface has the structure γ×S2, the words ‘contin-uous through the curve γ’ can be interchanged with thewords ‘continuous through the hypersurface Γ ’, depend-ing on ones preference to think four- or two-dimensional.A great advantage of the SSJC is that they are very

easy to verify: one simply has to impose continuity

11 The minus sign here is just a relict of our signature choice.12 For an English translation see (Lemaıtre, 1997).

15

on four scalars along the matching curve in the two-dimensional base manifold B. Dealing with scalars, sincetheir value is independent on the particular coordinatechoice, one does not need to worry about introducingnew coordinates in both spacetimes to be glued, in orderto get the different metrics in a form which is comparable.This is indeed an ingrate task: in general, these coordi-nates are only needed in order to check if the junctionconditions are satisfied, and for nothing more. In pres-ence of spherical symmetry all this can be circumventedby using our new junction conditions.Furthermore, the SSJC have a good physical interpre-

tation: The continuity condition of both, the areal radiusas well as of the MS energy, can be read as equilibriumcondition for the gravitational pull acting from oppositedirections onto (fictitious) test masses at the location ofthe matching surface. Concerning the continuity of theextrinsic curvature of the matching curve, we note thefollowing: In the case where the matching hypersurfaceΓ is timelike, let γ = π(Γ ) be a timelike curve in B andv is future-pointing tangent. One can think of γ as the‘matching observer’s’ worldline. Hence, in the timelikecase, the extrinsic curvature is nothing but the acceler-ation of the matching observer. In the spatial case, onthe contrary, n is timelike, γ is spacelike (and v tangentto it). One can choose n to be future-pointing and thenthink of it as an observer field defined along the spatial(1-dimensional) slice γ. In view of (C7) one then seesthat the extrinsic curvature of γ is exactly13 the shear-expansion of n in direction of v ‘radial direction’.In Appendix B we prove the following

Theorem 1 (Equivalence of the junction conditions).Let Γ be a smooth, non-null, spherically symmetricmatching hypersurface between two spherically symmet-ric spacetimes and n a continuous choice of unit normalvector field on Γ . Assume, moreover, that the areal radiiof the two spacetimes are C1 functions in an open neigh-borhood of the matching hypersurfaces. Then the DJCare equivalent to the SSJC.

Now let us suppose we are faced with the followingsituation: We are given two spherically symmetric solu-tions of Einstein equation and we want to know if theycan be matched together at all, and if so, how to char-acterize the curve γ (respectively, the hypersurface Γ )along which this is possible. Answers to these questionswill be provided by the junction conditions SSJC. Notethat a timelike or spacelike curve in the two-dimensionalbase manifold (B, gB) can be described simply by a func-tion R(τ), where R is the areal radius and τ the curve’sarc-length which, in the timelike case, corresponds to thematching observer’s proper time. The conditions (i) and(iii) of the SSJC are then equivalent to the condition

13 Recall that, because of our signature choice, the restriction ofthe metric to spacelike directions is negative definite.

that the functional dependence R(τ) must be the same(up to a trivial translation in τ) in both spacetimes to bematched.

B. The Eisenstaedt theorem

Perhaps the simplest attempt to model a compactbody (star) in an expanding universe is trying to inglo-bate it in a FLRW spacetime and to assume, for sim-plicity, that the body is spherically symmetric. A directconsequence of the SSJC is the following intuitive appeal-ing theorem due to Eisenstaedt (Eisenstaedt, 1977):

Theorem 2 (Eisenstaedt, 1977). Excide the full world-tube Wr0 of a comoving ball of comoving radius r0 from aFLRW spacetime and insert instead a spherical symmet-ric inhomogeneity (hence a piece of a spherically sym-metric spacetime together with a related matter model,satisfying Einstein’s equations). Then a necessary condi-tion for the resulting spacetime to satisfy Einstein’s equa-tion is that the MS energy of the inserted inhomogeneityequals that of the excided ball.

This says that the mean energy density (measured withthe MS energy) of spherically-symmetric inhomogeneitiesmust be the same as the one of the FLRW spacetime.That the Eisenstaedt Theorem is a consequence of theDarmois junction conditions was also seen in (Hartl,2006).

C. The Einstein–Straus vacuole revisited

As another application of the above described match-ing procedure we revisit the Einstein–Straus solu-tion (Einstein and Straus, 1945, 1946; Schucking, 1954),which originally consists on a Schwarzschild spacetime(called ‘vacuole’) matched to a dust FLRW universe withzero cosmological constant. Later, this model was gener-alized also to the case of a nonvanishing cosmological con-stant (Balbinot et al., 1988). We treat here the generalcase of an arbitrary cosmological constant and show thatthe SSJC allow substantial simplifications of the compu-tations. This technique can also be applied to Bonnor’svacuole construction in LTB spacetimes.

Notice that the matching condition (56) implies, inparticular, that the pressure must be continuous throughthe matching hypersurface. Since the interior is a vacuumspacetime, it follows that the pressure must vanishes alsoon the exterior part of the matching hypersurface andhence everywhere on the FLRW spacetime. That is whyone has to restrict to dust FLRW spacetimes.

Since we leave the cosmological constant Λ arbitrary(it may be positive, negative, or zero) the inner re-gion is given, respectively, by a Schwarzschild–deSitter,

16

Schwarzschild–anti-deSitter14, or Schwarzschild space-time (all abbreviated henceforth by SdS). Recall that theSdS spacetime is given by the vacuum solution to Ein-stein equation with cosmological term

gSdS = V (R)dT 2 − V (R)−1dR2 −R2gS2 , (59a)

where

V (R) = 1− 2m

R− Λ

3R2 . (59b)

Above, gS2 denotes the metric on the unit two-sphere (31) and m is a constant which represents thecentral mass.A dust FLRW spacetime is given by the metric

gFLRW = dt2 − a(t)2(

dr2

1− kr2+ r2gS2

)

(60)

together with the matter energy-momentum tensor15

T = u ⊗ u, where u = ∂/∂t is the (geodesic) velocityfield of the cosmological dust and is the matter energydensity, which depends on t only. Here r is the comov-ing radial coordinate and k is a constant which takes thevalues 0,−1,+1, depending on whether the spatial sliceshave zero, negative, or positive curvature, respectively.The Einstein equation is then equivalent to the followingset given by the Friedmann equation and a ‘conservationequation’:

(a

a

)2

+k

a2− C

a3− Λ

3= 0 , (61a)

a3 = constant, (61b)

where the constant C := 8π0a30/3 depends on the initial

conditions a0 := a(t0) and 0 := (t0) at some ‘initial’time t0. Here, the dot denotes differentiation with respectto t or, which is the same, along u.The central question is now the following: How shall we

cut hypersurfaces ΓSdS = γSdS ×S2 and ΓFLRW = γFLRW ×S2 in the spacetimes SdS and, respectively, FLRW inorder that the resulting pieces can be matched? In orderto apply the SSJC we have to compute the MS energyfor both spacetimes. For the FLRW spacetime one hasRFLRW(t, r) = a(t)r and hence dRFLRW = ardt+ adr and〈dR,dR〉FLRW = gµνR,µR,ν = r2(k + a2) − 1. From thedefinition (58) one then gets, using Friedmann’s equation,

EFLRW =4π

3R3

FLRW(+ Λ) , (62)

14 The Schwarzschild–(anti-) de Sitter metric (59a) is often calledthe Kottler solution, after Friedrich Kottler, who was the firstto write down this metric in (Kottler, 1918). More details on itsanalytic and global structure may be found in (Geyer, 1980).

15 Throughout we denote the metric-dual (1-form) of a vector u

by underlining it, that is, u := g(u, · ) is the 1-form metric-dualto vector u. In local coordinates we have u = uµ∂µ and u=uµdxµ, where uµ := gµνuν .

where Λ := Λ/8π is the energy density associated withthe cosmological constant. Notice that this expression,as well its derivation, is completely independent of thespecific equation of state of the fluid and do not dependon the spatial curvature k. For the SdS case one hasRSdS = R and thus 〈dR,dR〉SdS = −V (R) and

ESdS = m+4π

3R3

SdSΛ . (63)

Now, the last two conditions of the SSJC, that is thecontinuity of areal radius and MS energy, across thematching hypersurface (yet still to be determined) areequivalent to the continuity of the areal radius RFLRW =RSdS =: R, together with the suggestive relation

m =4π

3R3 . (64)

This two conditions already determine the matching hy-persurface: Inserting R = RFLRW = a(t)r in (64) and us-ing the dust FLRW relation (t)a3(t) = const. one getsthe matching radius expressed in the FLRW comovingradius:

r = constant =

(m

(4π/3)a300

)1/3

=: r0 (65)

Here a0 := a(t0), and similarly for , where t0 is somefixed ‘initial’ time. This means that the matching ob-server moves, in the FLRW spacetime, along the inte-gral curve of u = ∂/∂t with initial condition (t0, r0) andhence is comoving with the cosmological matter.So far we used the last two of the SSJC. As discussed

above, the continuity of the areal radius and the arc-length (the proper time, in the timelike case) of thematching curve are equivalent to the equality of the func-tional dependencies R(τ) (up to a possible trivial trans-lation in τ) which describe the matching curves in thetwo spacetimes to be matched. Now, because of (65),the matching curve (worldline) in the FLRW spacetimeis simply

R(τ) = a(τ)r0 , (66)

where a is the (unique) solution of the Friedmann equa-tion (61a) with initial condition a0 at τ0 = t0. (Recallthat in FLRW the proper time of an observer movingalong an integral line of u equals the cosmological time,hence τ = t.) From what we said above, the samefunctional relation R(τ) must hold also in the SdS—provided we identify R with the areal radius and τ withthe matching observer’s proper time, both referred to theSdS spacetime. This determines the matching curve inSdS.Finally we need to show that the junction condition (ii)

is satisfied, hence that the matching observer’s accelera-tions coincide. Looking at the matching worldline fromthe FLRW spacetime, it is immediately clear that it isgeodesic, hence its acceleration vanishes. To conclude

17

the matching procedure, we just have to check that thisis also true for the matching worldline in the SdS space-time. For this, one has just to check that the functiondefined in (66) satisfies the geodesic equation for a radialmotion. The latter is given by

R2 + V (R) = e2 , (67)

where e := gSdS(∂/∂T,v) = constant and v = T ∂/∂T +

R ∂/∂R is the matching observer in the SdS space-time. (Equation (67) can be quickly derived from the factthat v(gSdS(∂/∂T,v)) = 0, since ∂/∂T is Killing and

v geodesic. Inserting e := gSdS(∂/∂T,v) = V (R)T in

the normalization condition 1 = gSdS(v,v) = V (R)T 2 −R2/V (R) one arrives immediately at (67).) Now, insert-ing (66) with (65) in (67) and using the Friedmann equa-

tion (61a), one gets R2 + V (R) = 1 − kr20. Hence, the

geodesic equation (67) is satisfied (with e2 = 1 − kr20)

and herewith all the four junction conditions.

D. The McVittie model

Among all models discussed in the literature which rep-resent a quasi-isolated spherically-symmetric gravitatingsystem melted into a cosmological spacetime, the onethat is presumably best understood as regards its an-alytical structure as well as its physical assumptions isthat of McVittie (McVittie, 1933), thanks to the carefulanalysis of Nolan (Nolan, 1998, 1999a,b). Here we shallrestrict to the ‘flat’ or k = 0 model, which interpolatesbetween an exterior Schwarzschild solution, describing alocal mass, and a spatially flat (i.e. k = 0) ambient FLRWuniverse. For simplicity we shall from now on refer to thismodel simply as the McVittie model. The cosmologicalconstant is assumed to be zero, although this assumptionis not essential (see the last paragraph of Section V.D.1).This is not to say that this model is to be taken at

face value in all its aspects. Its problems lie in the re-gion very close to the central object, where the basicassumptions on the behavior of matter definitely turnunphysical. However, at radii much larger than (in geo-metric units) the central mass (to be defined below) thek = 0 McVittie solution seems to provide a viable ap-proximation for the transition between a homogeneouscosmological spacetime and a localized mass immersedin it. We will now briefly discuss this model and look atits geodesic equations, showing that they reduces to (12)in an appropriate weak-field and slow-motion limit. Thisprovides another and more solid justification for the New-tonian approach we carried out in Section III.The characterization of the McVittie model is made

through two sets of a priori specifications. The first setconcerns the metric (left side of Einstein’s equations) andthe second set the matter (right side of Einstein’s equa-tions). The former consists in an ansatz for the metric,which can formally be described as follows: Write downthe Schwarzschild metric for the mass parameter m in

isotropic coordinates, add a conformal factor a2(t) to thespatial part, and allow the mass parameter m to dependon time. Hence the metric reads

g =

(1−m(t)/2r

1 +m(t)/2r

)2

dt2

−(

1 +m(t)

2r

)4

a2(t) (dr2 + r2gS2) ,

(68)

where gS2 is given by (31). The metric (68) is obviouslyspherically symmetric with the spheres of constant ra-dius r being the orbits of the rotation group. We willdiscuss below what this ansatz actually entails. For laterconvenience we also introduce the orthonormal tetrad{eµ}µ=0,··· ,3 with respect to (68), where16

eµ := ‖∂/∂xµ‖−1 ∂/∂xµ (69)

and {xµ} = {t, r, θ, ϕ}.The second set of specifications, concerning the matter,

is as follows: The matter is an ideal fluid with density and isotropic pressure p. Hence its energy-momentumtensor is given by

T = u⊗ u+ p (u⊗ u− g) . (70)

Furthermore, and this is where the two sets of specifi-cations make contact, the motion of the matter is givenby

u = e0 . (71)

No further assumptions are made. In particular, an equa-tion of state, like p = p(), is not assumed. The reasonfor this will become clear soon.Note that the vector field (71) is not geodesic for the

metric (68) (unlike for the FLRW and Gautreau metrics),which immediately implies that the pressure cannot beconstant. Being spherically symmetric, u is automati-cally vorticity free. The last property is manifest fromits hypersurface orthogonality, which is immediate from(68). Moreover, u is also shear free. This, too, can beimmediately read off (68) once one takes into accountthe result that for spherically symmetric metrics van-ishing shear for a spherically symmetric vector field isequivalent to the corresponding spatial metric being con-formally related to a spherically symmetric flat metric.This is obviously the case here.The nonvanishing components of the Einstein tensor

with respect to the orthonormal basis (69) are:

Ein(e0, e0) = 3F 2 , (72a)

Ein(ei, ej) = −(

3F 2 + 2AB F)

δij , (72b)

Ein(e0, e1) =2R2

(AB

)2(am)˙ , (72c)

16 We write ‖v‖ :=p

|g(v, v)|.

18

where an overdot denotes differentiation along ∂/∂t. Be-fore explaining the functions A, B, R, and F , we makethe important observation that the Einstein tensor is spa-tially isotropic, where ‘spatially’ refers to the directionsorthogonal to e0. By this we mean that Ein(ei, ej) ∝ δijor, expressed more geometrically, that the spatial restric-tion of the Einstein tensor is proportional to the spatialrestriction of the metric.In (72) and in the following we set:

A(t, r) := 1 +m(t)/2r , B(t, r) := 1−m(t)/2r , (73)

and

R(t, r) =

(

1 +m(t)

2r

)2

a(t) r , (74)

where R is the areal radius for the McVittie ansatz (68),and also

F :=a

a+

1

rB

(am)·

a. (75)

In passing we note that F has the geometric interpreta-tion of being one third the expansion of the vector fielde0, that is, F = div(e0)/3. Hence (72a) could also bewritten in the form Ein(e0, e0) = (div(e0))

2/3. We willsee later that the product am which appears in (72c)also has a geometric meaning: it is just the Weyl part ofthe MS energy; see (81).Now, the nonvanishing components of the energy mo-

mentum tensor (70) with (71) are:

T (e0, e0) = , T (ei, ej) = p δij . (76)

The (e0, e1) component of Einstein’s equation therefore

implies (am)˙ = 0, which means that the Weyl part of theMS energy is constant. Physically this can be interpretedas saying that the central object does not accrete anyenergy from the ambient matter. Using the constancy ofam in (75) we immediately get:

F =a

a=: H . (77)

Hence Einstein’s equation is equivalent to the fol-lowing three relations between the four functionsm(t), a(t), (t, r), and p(t, r):

(am)˙ = 0 , (78a)

8π = 3

(a

a

)2

, (78b)

8πp = −3

(a

a

)2

− 2

(a

a

)˙ (1 +m/2r

1−m/2r

)

. (78c)

Note that here Einstein’s equation has only three inde-pendent components (as opposed to four for a generalspherically symmetric metric), which is a consequence ofthe fact, already stresses above, that the Einstein tensorfor the McVittie ansatz (68) is spatially isotropic.

Equation (78a) can be immediately integrated:

m(t) =m0

a(t), (79)

where m0 is an integration constant. Below we will showthat this integration constant is to be interpreted as themass of the central body. We will call the metric (68)together with condition (79) the McVittie metric.Clearly the system (78) is under-determining. This is

expected since no equation of state has yet been imposed.The reason why we did not impose such a condition cannow be easily inferred from (78): whereas (78b) impliesthat only depends on t, (78c) implies that p dependson t and r iff (a/a) 6= 0. Hence a non-trivial relationp = p() is simply incompatible with the assumptionsmade so far. The only possible ways to specify p arep = 0 or + p = 0. In the first case (78c) implies thata/a = 0 if m0 6= 0 (since then the second term on theright-hand side is r dependent, whereas the first is not,so that both must vanish separately), which correspondsto the exterior Schwarzschild solution, or a(t) ∝ t2/3 ifm0 = 0, which leads to the flat FLRW solution withdust. In the second case the fluid just acts like a cos-mological constant Λ = 8π (using the equation of state + p = 0 in divT = 0 it implies dp = 0 and this, inturn, using again the equation of state, implies d = 0)so that this case reduces to the Schwarzschild–deSittersolution. To see this explicitly, notice first that (78b,78c)

imply the constancy of H = a/a =√

Λ/3 and hence

one has a(t) = a0 exp(t√

Λ/3). With such a scale-

factor the McVittie metric (68) with (79) turns into theSchwarzschild–deSitter metric (59) in disguise. The ex-plicit formulae for the coordinate transformation relatingthe two can be found in Section 5 of (Robertson, 1928)and also in Section 7 of (Klioner and Soffel, 2005). Fi-nally, note from (78a) that constancy of one of the func-tions m and a implies constancy of the other. In this case(78b,78a) imply p = = 0, so that we are dealing withthe exterior Schwarzschild spacetime.A specific McVittie solution can be obtained by choos-

ing a function a(t), corresponding to the scale functionof the FLRW spacetime which the McVittie model isrequired to approach at spatial infinity, and the con-stant m0, corresponding to the ‘central mass’. Rela-tions (78b,78c), and (79) are then used to determine ,p, and m, respectively. Clearly this ‘poor man’s way’to solve Einstein’s equation holds the danger of arrivingat unrealistic spacetime dependent relations between and p. This must be kept in mind when proceeding inthis fashion. For further discussion of this point we referto (Nolan, 1998, 1999a).

1. Interpretation of the McVittie model

In this section we discuss the interpretation of theMcVittie model, its singularities, trapped regions, sym-metry properties, and also the motion of the matter. In

19

doing this, we shall take care to isolate those propertieswhich are intrinsic to the ansatz (68) independent of theimposition of Einstein’s equation. The analysis can thenalso be applied to all generalizations which maintain theansatz (68). Generalizations in this sense have recentlybeen discussed in the literature (Faraoni and Jacques,2007; Sultana and Dyer, 2005), on which we will com-ment at the end of this section.According to what has just been said, we wish to re-

gard the McVittie solution as candidate model for anisolated mass m0 in an ‘otherwise’ flat FLRW universewith scale function a(t). As already emphasized in theintroduction, this requires specific justification in view ofthe fact that simple superpositions of solutions are dis-allowed by nonlinearities. A set of criteria for when asolution represents a localized mass immersed in a flatFLRW background have been proposed and discussed indetail in (Nolan, 1998). The basic idea is to employ theMS energy (in a spherically symmetric context, where itis equivalent to the Hawking mass) in order to detect lo-calized sources of gravity. We will follow this approachand for this purpose we compute the Ricci and the Weylpart of the MS energy.This we now do for the class of metrics (68), with-

out at first making any use of Einstein’s equation. Thegeometric definition of the MS energy in terms of thesectional curvature, together with formula (A10b) for itsRicci part specialized to metrics with spatially isotropicEinstein tensor, implies that the Ricci part of the MSenergy is given by

ER = 16R

3 Ein(e0, e0) . (80)

The Weyl part is then obtained as the difference betweenthe full MS energy and (80). We use the expression (58)

for the former and write 〈dR,dR〉 =(e0(R)

)2−(e1(R)

)2.

The part involving e0(R) equals (80), due to the relationEin(e0, e0) = 3(dR(e0)/R)2, which, e.g., follows fromthe comment below Eq. (75) and (D52) (for vanishingshear). The Weyl part of the MS energy is therefore givenby (R/2)

(1 − (e1(R))2

). From (74) we calculate e1(R)

and hence obtain for the Weyl part of the MS energy:

EW = am . (81)

Now we invoke Einstein’s equation with source (70)and four-velocity (71). Then the Ricci and Weyl contri-butions to the MS energy can be written in the followingform, also taking into account (79),

ER =4π

3R3 , (82a)

EW = m0 . (82b)

Identifying the gravitational mass of the central objectwith the Weyl part of the MS energy, its constancy meansthat no energy is accreted from the ambient matter. Asregards the Ricci part, note that the factor (4π/3)R3

in (82a) is smaller than the proper geometric volume

within the sphere of areal radius R. This can be at-tributed to the gravitational binding energy that dimin-ishes the gravitational mass of a lump of matter belowthe value given by the proper space integral of T (e0, e0).This is shown in more detail in Appendix D.4, in par-ticular in the exact equation (D45) and its leading orderapproximation (D47).It is also important to note that the central gravita-

tional mass in McVittie’s spacetime may be modeled by ashear-free perfect-fluid star of positive homogeneous en-ergy density (Nolan, 1993). The matching is performedalong a world-tube comoving with the cosmological fluid,across which the energy density jumps discontinuously.This means that the star’s surface is comoving with thecosmological fluid and hence, in view of (89), that it ge-ometrically expands (or contracts). This feature, how-ever, should be merely seen as an artifact of the McVittiemodel (in which the relation (89) holds), rather than ageneral property of compact objects in any cosmologicalspacetimes. Positive pressure within the star seems to beonly possible if 2aa + a2 < 0 (see Eq. (3.27) in (Nolan,1993) with a = exp(β/2)), that is, for deceleration pa-rameters q > 1/2.Next we comment on the singularity properties of the

McVittie model. From (78c) it is clear that, unless a/ais constant (the Schwarzschild–deSitter case) or m = 0(FLRW case), the pressure diverges at r = m/2 (that isat R = 2m0 = RS). In fact, this corresponds to a gen-uine curvature singularity which is built into the McVit-tie ansatz (68) independently of any further assumption.To see that r = m/2 (corresponding to R = 2am = 2EW)is a singularity it suffices to consider the scalar curvature(i.e. the Ricci scalar) of (68),

Scal = −12F 2 − 6AB F , (83)

which is readily computed from (72).In (Carrera and Giulini, 2008) we show that thisbecomes singular in the limit r → m/2, with theonly exceptions being the following three specialcases: (i) m = 0 and a arbitrary (FLRW spacetimes),(ii) a and m constant (Schwarzschild spacetime), and(iii) (am)· = 0 and (a/a)· = 0 (Schwarzschild–deSitterspacetime). This means that, as long as we stick to theansatz (68), at r = m/2 there will always (with the onlyexceptions listed above) be a singularity in the Riccipart of the curvature and thus, assuming Einstein’sequation is satisfied, also in the energy momentumtensor, irrespectively of the details of its underlyingmatter model. Hence any attempt to eliminate thissingularity by maintaining the ansatz (68) and merelymodifying the matter model is doomed to fail. Inparticular, this is true for the generalizations presentedin (Faraoni and Jacques, 2007), contrary to what isclaimed in that work. We also remark that it makesno sense to absorb the singular factors 1/B in front ofthe time derivatives by writing (A/B)∂/∂t as e0 andthen argue, as was done in (Faraoni and Jacques, 2007),that this eliminates the singularity. The point is simply

20

that then e0 applied to any continuously differentiablefunction diverges as r → m/2. Below we will show thatthis singularity lies within a trapped region. Turningback to the McVittie model, recall that in this case it isassumed that the fluid moves along the integral curvesof ∂/∂t, which become lightlike in the limit as r tendsto m/2. Their acceleration is given by the gradientof the pressure, which necessarily diverges in the limitr → m/2, as one explicitly sees from (91). For a moredetailed study of the geometric singularity at r = m/2,see (Nolan, 1999a,b).For spherically symmetric spacetimes the Weyl part of

the curvature has only a single independent component(see (D8a)) which, by its very definition, is −2/R3 timesthe Weyl part of the MS energy (see (D36)). The squareof the Weyl tensor for the ansatz (68) may then be con-veniently expressed as (see (81) and (D37))

〈Weyl,Weyl〉 = 48(am)2

R6. (84)

This shows that R = 0 also corresponds to a genuinecurvature singularity, though this is not part of the regioncovered by our original coordinate system, for which r >m/2 (that is R > 2EW).It is instructive to also determine the trapped regions

of McVittie spacetime. We do this just using the McVit-tie ansatz (68) and making no further assumptions. Re-call that a spacelike two-sphere S is said to be trapped,marginally trapped, or untrapped if the product θ+θ− ofthe expansions (defined below Eq. (D29)) for the ingo-ing and outgoing future-pointing null vector fields nor-mal to S is positive, zero, or negative. Taking S to beSR, that is, a sphere of symmetry with areal radius R,it immediately follows from (D31) that SR is trapped,marginally trapped, or untrapped iff 〈dR,dR〉 is posi-tive, zero, or negative, respectively. This corresponds totimelike, lightlike, or spacelike dR, or, equivalently, inin view of (58), to 2E − R being positive, zero, or neg-ative, respectively. Using (80) together with (72a), theMS energy for the McVittie ansatz can be written asE = EW +R3F 2/2, so that

R2H(2E −R) = R3 −R2

HR+R2HRS , (85)

where we defined the ‘generalized’ Hubble- andSchwarzschild radius as RH := 1/F and RS := 2EW.This is a cubic polynomial in R which is positive forR = 0 and tends to ±∞ for R → ±∞. Hence it alwayshas a negative zero (which does not interest us) and twopositive zeros iff

RS/RH < 2/3√3 ≈ 0.38 . (86)

This clearly corresponds to the physical relevant casewhere the Schwarzschild radius is much smaller thanthe Hubble radius. One zero lies in the vicinity ofthe Schwarzschild radius and one in the vicinity of theHubble radius, corresponding to two marginally trappedspheres. The exact expressions for the zeros can be easily

written down, but are not very illuminating. In leadingorder in the small parameter RS/RH , they are approxi-mated by

R1 ≈ RS

(1 + (RS/RH)2

), (87a)

R2 ≈ RH (1−RS/2RH) . (87b)

From this one sees that for the McVittie ansatz the radiusof the marginally trapped sphere of Schwarzschild space-time (RS) increases and that of the FLRW spacetime(RH) decreases. The first feature can, for the McVittiemodel, be understood as an effect of the presence of cos-mological matter, whereas the latter is an effect of thepresence of a central mass abundance. All the sphereswith R < R1 or R > R2 are trapped and those withR1 < R < R2 are untrapped. In particular, the singu-larity r = m/2, that is R = 2EW = RS , lies within theinner trapped region.Another aspect concerns the global behavior of the

McVittie ansatz (68). We note that each hypersurfaceof constant time t is a complete Riemannian manifold,which, besides the rotational symmetry, admits a dis-crete isometry given in (r, θ, ϕ) coordinates by

φ(r, θ, ϕ) =((m/2)2 r−1 , θ , ϕ

). (88)

This corresponds to an inversion at the two-sphere r =m/2 and shows that the hypersurfaces of constant t canbe thought of as two isometric asymptotically-flat piecesjoined together at the totally geodesic (being a fixed-point set of an isometry and hence also minimal) two-sphere r = m/2. Except for the time-dependent fac-tor m(t), this is just like for the slices of constant t inthe Schwarzschild metric (the difference being that (88)does not extend to an isometry of the spacetime met-ric unless m = 0). This means, in particular, that theMcVittie metric cannot literally be interpreted as cor-responding to a point particle sitting at r = 0 (r = 0is in infinite metric distance) in a flat FLRW universe,just like the Schwarzschild metric does not correspondto a point particle sitting at r = 0 in Minkowski space.Unfortunately, McVittie seems to have interpreted hissolution in this fashion (McVittie, 1933) which even un-til recently gave rise to some confusion in the literature(e.g. (Ferraris et al., 1996; Gautreau, 1984; Sussman,1988)). A clarification was given by (Nolan, 1999a).We now briefly discuss the basic properties of the mo-

tion of cosmological matter for the McVittie model. Wealready mentioned that the vorticity and shear of thefour-velocity u vanish identically. On the other hand,the expansion (divergence of u) is

θ = 3H , (89)

just as in the FLRW case (recall that here H := a/a isdefined as in the FLRW case, see (77)). In particular,the expansion of the cosmological fluid is homogeneousin space. Exactly as in the FLRW case is also the ex-pression for the variation of the areal radius along the

21

integral lines of u (that is the velocity of cosmologicalmatter measured in terms of its proper time and the arealradius):

u(R) = HR , (90)

which is nothing but Hubble’s law. The acceleration ofu, which in contrast to the FLRW case does not vanish,is given by

∇uu =m0

R2

(1 +m/2r

1−m/2r

)

e1 . (91)

In leading order in m0/R this corresponds to theacceleration of the observers moving along ∂/∂T inSchwarzschild spacetime (see Eq. (134) with Λ = 0).We conclude this subsection commenting the attempts

to generalize the McVittie model. The first obvious gen-eralization consists in allowing a nonvanishing cosmologi-cal constant. This is however trivial, since it is equivalentto the substitution → + Λ and p → p + pΛ in (78),where Λ := Λ/8π and pΛ := −Λ/8π are, respectively,the energy-density and pressure associated to the cosmo-logical constant Λ. In the literature, attempts to general-ize the McVittie model have have focused on keeping theansatz (68) and relaxing the conditions on the matter invarious ways. In (Faraoni and Jacques, 2007) generaliza-tion were presented allowing radial fluid motions relativeto the e0 observer field (that is relaxing condition (71))as well as including heat conduction. Another solutionwhich can be seen as a generalization of the McVittiemodel, but was originally derived by a particular con-formal transformation of Schwarzschild spacetime, canbe found in (Sultana and Dyer, 2005). In this model, tothe perfect fluid one incoherently adds (i.e. the energy-momentum tensors add) another matter component cor-responding to a radially incoming null fluid. Its energy-momentum tensor is nk⊗k, where k the radially inwardpointing null vector. Then one may take a(t) ∝ t2/3 if(79) is replaced with m = const. This means that theWeyl part of the MS energy (equal to the Weyl part of theHawking mass) grows in proportion to a(t). For an anal-ysis of these solutions see (Carrera and Giulini, 2008).

2. Motion of a test particle in McVittie spacetime

We are interested in the motion of a test particle (ide-alizing a planet or a spacecraft) in McVittie’s space-time. In (McVittie, 1933) it was concluded within aslow-motion and weak-field approximation that Keple-rian orbits do not expand as measured with the ‘cos-mological geodesic radius’ r∗ = a(t)r. Later (Pachner,1963) and (Noerdlinger and Petrosian, 1971) argued forthe presence of the acceleration term (7) proportionalto a/a within this approximation scheme, hence arrivingat (12a). In the following we shall show how to arrive at(12a) from the exact geodesic equation of the McVittiemetric by making clear the approximations involved. Re-lated recent discussions were given in (Bolen et al., 2001),

where the effects of cosmological expansion on the peri-astron precession and eccentricity are discussed for con-stant Hubble parameter H := a/a.We will again work with the areal radius R. Note

that for fixed t the map r 7→ R(t, r) is 2-to-1 and thatR ≥ 2m0, where R = 2m0 corresponds to r = m0/2a.Hence we restrict the coordinate transformation (74) tothe region r > m0/2a where it becomes a diffeomorphismonto the region R > 2m0. (The region R < 2m0 was in-vestigated in (Nolan, 1999b).) Reintroducing factors ofc, McVittie’s metric assumes the (non-diagonal) form inthe region R > 2m0 (i.e. r > m0/2a(t))

g =(1− 2µ(R)− h(t, R)2

)c2dt2

+2h(t, R)

√

1− 2µ(R)cdtdR− dR2

1− 2µ(R)−R2gS2 ,

(92)

where we put

µ(R) :=m0

R, h(t, R) :=

H(t)R

c(93)

with H(t) := (a/a)(t), as usual.The equations for a timelike geodesic (i.e. parame-

terized with respect to proper time), τ 7→ zµ(τ) withg(z, z) = c2, follows via variational principle from theLagrangian L(z, z) = (1/2)gµν(z)z

µzν . Spherical sym-metry implies conservation of angular momentum. Hencewe may choose the particle’s orbit to lie in the equato-rial plane θ = π/2. The constant modulus of angularmomentum is

R2ϕ = L . (94)

The remaining two equations are then coupled second-order ODEs for t(τ) and R(τ). However, we may re-place the first one by its first integral that results fromg(z, z) = c2:

(1− 2µ(R)− h2(t, R)

)c2t2

+2h(t, R)

√

1− 2µ(R)c t R− R2

1− 2µ(R)− (L/R)2 = c2 .

(95)

The remaining radial equation is given by

R −(1− 2µ(R)− h2(t, R)

) L2

R3

+m0 c

2

R2

(1− 2µ(R)

)t2

−R(

H(t)(1− 2µ(R)

)1/2

+H(t)2(1− µ(R)− h2(t, R)

) )

t2

−(µ(R)− h2(t, R)

)

1− 2µ(R)

R

R

2

+2(µ(R)− h2(t, R)

)

√

1− 2µ(R)cH(t) (R/c) t = 0 .

(96)

22

Recall that m0 = GM/c2, where M is the mass of thecentral star in standard units.Equations (95,96) are exact. We are interested in or-

bits of slow-motion (compared with the speed of light) inthe region where

RS ≪ R ≪ RH . (97)

Recall that RS and RH are the Schwarzschild and theHubble radius, respectively (see (5),(6)). The latter con-dition clearly covers all situations of practical applica-bility in the Solar System, since the Schwarzschild radiusRS of the Sun is about 3 km = 2·10−8AU and the Hubbleradius RH is about 13.7 · 109 ly = 8.7 · 1014AU.The approximation now consists in considering small

perturbations of Keplerian orbits. Let T be a typical timescale of the problem, like the period for closed orbits orelse R/v with v a typical velocity. The expansion is thenwith respect to the following two parameters:

ε1 ≈v

c≈(m0

R

) 12

, (98a)

corresponding to a slow-motion and weak-field approxi-mation, and

ε2 ≈ HT , (98b)

corresponding to the approximation for small ratios ofcharacteristic-times to the age of the universe. In order tomake the expression to be approximated dimensionless,we multiply (95) by 1/c2 and (96) by T 2/R. Then weexpand the right hand sides in powers of the parameters(98), using the fact that h := (HR/c) ≈ ε1ε2. From thisand (94) we obtain (12) if we keep only terms to zero-order in ε1 and leading (i.e. quadratic) order in ε2, wherewe also re-express R as function of t. Note that in thisapproximation the areal radius R is equal to the spatialgeodesic distance on the t = const. hypersurfaces.We already mentioned that in the special case of con-

stant H = a/a the McVittie solution turns into theSchwarzschild–deSitter metric (59). In this case thegeodesic equation can be integrated exactly in termsof hyper-elliptic integrals (Hackmann and Lammerzahl,2008a,b). A general discussion of Solar-System effects inSchwarzschild–deSitter, like gravitational redshift, lightdeflection, time delay, perihelion precession, geodetic pre-cession, and effects on Doppler tracking, has been givenin (Kagramanova et al., 2006). For example, it was foundthat a nonvanishing Λ could account for the anomalousPioneer acceleration if its value was −10−37m−2, whichis minus 1015 times the current most probable value, andalso would give rise to a perihelion precession four ordersof magnitude larger than the accuracy to which this effecthas been measured.We conclude by commenting on the geodesic

equation in the generalizations of McVittie’smodel given in (Faraoni and Jacques, 2007) andin (Sultana and Dyer, 2005). The essential featurewhich distinguish these solutions from the McVittie one,

is that in the former the Weyl part of the MS energyEW = am is not a constant as for McVittie but varies intime, meaning that there is an accretion of cosmologicalmatter by the inhomogeneity (see (Carrera and Giulini,2008)). In view of the fact that the combination

m/r = A2EW/R ≈ EW/R (99)

contained in the McVittie ansatz gives (minus) the ‘New-tonian’ part of the potential in the slow-motion and weak-field approximation (see Section V.D.2), we deduce thatin order to get the geodesic equation for the generalizedMcVittie models it suffices to substitute m0 with EW inthe equation of motion derived in Section V.D.2. Thismeans that the strength of the central attraction variesin time, leading to an in- or out-spiraling of the orbits ifEW is increasing or decreasing, respectively.

3. Exact condition for non-expanding circular orbits in McVittiespacetime

Analogously to Section IV.B, where we ask whetherthere exist non-expanding circular orbits (i.e. of con-stant areal radius) of the electron-proton system in aFLRW spacetime, we now ask whether there exist non-expanding circular orbits of an (uncharged) test parti-cle around the central mass. The necessary and suffi-cient condition for this to happen follows from insert-ing R = const in the radial part of the geodesic equa-tion (96) and using the normalization condition (95) ofthe four-velocity in order to eliminate t. In terms of thedimensionless quantities h(t) := RH(t)/c, l := L/Rc,and µ := m0/R, the condition for the existence of non-expanding circular orbits can be given the following form:

R

ch =

(1− 2µ− h2

) (µ(1 + 3l2)− l2 − h2

)

(1 + l2)√1− 2µ

. (100)

As for the electron-proton system in an FLRW space-time (see (51)) this is a first-order autonomous ODE forh(t) and therefore the Hubble function. In the presentcase the ODE is even simpler since it has the elementaryform h = p(h2), where p is a polynomial of degree twowith constant coefficients. From (100), to leading orderin the small quantities µ, l2, and h2, we get the sameapproximate ODE (54) and hence the same approximatesolutions (55). Hence, the same conclusions as drawn forthe electron-proton system in FLRW apply here.From (100) it follows that stationary solutions h(t) =

const =: h0, corresponding to an exponentially-growingscale factor (52) (and hence leading to a Schwarzschild–de Sitter spacetime), are those where h0 satisfies

l2 + h20

(1 + 3l2)= µ , (101)

where we used that the first factor on the numerator ofthe right-hand side of (100) is nonzero, as can be imme-diately inferred from the normalization condition (95).

23

Notice that for a vanishing Hubble parameter (that isfor h0 = 0) the above condition reduces to the third Ke-pler law in Schwarzschild spacetime, as expected. The ef-fect of a nonvanishing Hubble parameter is again that wemust provide the orbiting particle with a smaller angularvelocity (smaller l) in order to keep it on a non-expandingcircular orbit with the same radius. The largest radiusat which in a McVittie spacetime with exponentially-expanding scale factor (that is a Schwarzschild–deSitterspacetime) there is a non-expanding circular orbit followsfrom (101) in the limit l → 0. Then the condition reducesto h2

0 = µ which, solving for R, gives (RSR2H/2)1/3. This

exactly corresponds to the critical radius (16), taking intoaccount that q0 = −1 for an exponentially-growing scalefactor.

VI. KINEMATICAL EFFECTS

In this section we discuss the influence of cosmic ex-pansion upon measurements of relative distances, veloc-ities, and accelerations. These kinematical notions loosetheir a priori meaning in general spacetimes, in particu-lar in time-dependent ones. Hence it is of upmost impor-tance to carefully reconsider statements concerning suchnotions and their precise relations to locally observablequantities.

A. Einstein- versus cosmological simultaneity

Misidentifications in the notion of simultaneity cangive rise to apparent anomalies in velocities and ac-celeration. Such an effect has e.g. been suggestedin (Rosales and Sanchez-Gomez, 1998) and again in(Rosales, 2002) to be able to account for the PA. Theirargument says that in a spatially flat FLRW universethe mismatch between adapted cosmological coordinateson the one hand and radar coordinates on the other justamount to an apparent difference in radial acceleration ofmagnitude (1). We agree on the existence and conceptualimportance of such an effect but we disagree on the mag-nitude, which seems to have been grossly overestimatedas we will show below.The cause of such effects lies in the way one actually

measures spatial distances and determines the clock read-ings they are functions of (a trajectory is a ‘distance’ foreach given ‘time’). The point is this: equations of mo-tions give us, for example, simultaneous (with respectto cosmological time) spatial geodesic distances as func-tions of cosmological time. This is what we implicitlydid in the Newtonian analysis. But, in fact, spacecraftranging is done by exchanging electromagnetic signals.The notions of spatial distance and simultaneity therebyimplicitly used are not the same as those we referred toabove. Hence the analytical expression of the ‘trajectory’so measured will be different.We first recall the local version of Einstein simultaneity

in general spacetimes (M, g). We take ds = gµνdxµdxν

to carry the unit of length so that dτ = ds/c carriesthe unit of time. In general coordinates {xµ} = {t, xi},where x0 = t denotes the timelike coordinate, the metricreads

ds2 = gµνdxµ dxν = gttdt

2 + 2gtidt dxi + gijdx

i dxj .(102)

The observer at fixed spatial coordinates is given by thevector field (normalized to g(u,u) = c2)

u = c ‖∂/∂t‖−1∂/∂t =c√gtt

∂/∂t . (103)

Consider the light cone with vertex p ∈ M; one has ds2 =0, which allows to solve for dt in terms of the dxi (allfunctions gµν are evaluated at p, unless noted otherwise):

dt1,2 = − gtigtt

dxi ±√(gtigtjg2tt

− gijgtt

)

dxi dxj . (104)

The plus sign corresponds to the future light-cone at p,the negative sign to the past light cone. An integral lineof u in a neighborhood of p cuts the light cone in twopoints, q+ and q−. If tp is the time assigned to p, thentq+ = tp + dt1 and tq− = tp + dt2. The coordinate-timeseparation between these two cuts is tq+−tq− = dt1−dt2,corresponding to a proper time

√gtt(dt1 − dt2)/c for the

observer u. This observer will associate a radar-distancedl∗ to the event p of c/2 times that proper time interval,that is:

dl2∗ = h =

(gtigtjgtt

− gij

)

dxi dxj . (105)

The event on the integral line of u that the observerwill call Einstein-synchronous with p lies in the middlebetween q+ and q−. Its time coordinate is in first-orderapproximation given by 1

2 (tq++tq−) = tp+12 (dt1+dt2) =

tp + dt, where

dt := 12 (dt1 + dt2) = −gti

gttdxi . (106)

This means the following: The integral lines of uare parameterized by the spatial coordinates {xi}i=1,2,3.Given a point p, specified by the orbit-coordinates xi

p andthe time-coordinate tp, we consider a neighboring orbitof u with orbit-coordinates xi

p + dxi. The event on thelatter which is Einstein synchronous with p has a timecoordinate tp + dt, where dt is given by (106), or equiva-lently

θ := dt+gtigtt

dxi = 0 . (107)

Using a differential geometric language we may say thatEinstein simultaneity defines a distribution θ = 0.The metric (102) can be written in terms of the radar-

distance metric h (105) and the simultaneity 1-form θ asfollows:

ds2 = gµνdxµ dxν = gtt θ

2 − h , (108)

24

showing that the radar-distance is just the same as theEinstein-simultaneous distance. A curve γ in M inter-sects the flow lines of u perpendicularly iff θ(γ) = 0,which is just the condition that neighboring clocks alongγ are Einstein synchronized.We now apply the foregoing to isotropic cosmological

metrics. In what follows we drop for simplicity the an-gular dimensions. Hence we consider metrics of the form

ds2 = c2dt2 − a(t)2dr2 . (109)

The comoving observer field,

u = c∂/∂t , (110)

is geodesic and of expansion 3H . On a hypersurface ofconstant t the radial geodesic distance is given by a(t)r.Making this distance into a spatial coordinate, r∗, weconsider the coordinate transformation

t 7→ t∗ := t , r 7→ r∗ := a(t)r . (111)

The field ∂/∂t∗ is given by

∂/∂t∗ = ∂/∂t−Hr∂/∂r , (112)

to which the observer field,

u∗ := c ‖∂/∂t∗‖−1 ∂/∂t∗ (113)

corresponds. In contrast to (110), whose flow connectscomoving points of constant coordinate r, the flow of(112) connects points of constant geodesic distances,as measured in the surfaces of constant cosmologicaltime. This could be called cosmologically instantaneousgeodesic distance. It is now very important to realize thatthis notion of distance is not the same as the radar dis-tance that one determines by exchanging light signals inthe usual (Einsteinian) way. Let us explain this in detail:From (111) we have adr = dr∗ − r∗Hdt, where H :=

a/a (Hubble parameter). Rewriting the metric (109) interms of t∗ and r∗ yields

ds2 = c2(1− (Hr∗/c)2) dt2∗ − dr2∗ + 2Hr∗ dt dr∗

= c2{

1− (Hr∗/c)2}

︸ ︷︷ ︸

gt∗t∗

{

dt∗ +Hr∗/c

2

1− (Hr∗/c)2dr∗

︸ ︷︷ ︸

θ

}2

− dr2∗1− (Hr∗/c)2︸ ︷︷ ︸

h

. (114)

Hence the differentials of radar-distance and time-lapsefor Einstein-simultaneity are given by

dl∗ =dr∗

√

1− (Hr∗/c)2, (115a)

dt∗ = − Hr∗/c2

1− (Hr∗/c)2dr∗ . (115b)

Let the distinguished observer (us on earth) now movealong the geodesic r∗ = 0. Integration of (115) from r∗ =0 to some value r∗ then gives the radar distance l∗ as wellas the time lapse ∆t∗ as functions of the cosmologicallysimultaneous geodesic distance r∗:

l∗ = (c/H) sin−1(H r∗/c)

≈ r∗{1 + 1

6 (Hr∗/c)2 +O(3)

}(116a)

∆t∗ = (1/2H) ln(1− (H r∗/c)

2)

≈ (r∗/c){− 1

2 (Hr∗/c) +O(2)}

(116b)

Combining both equations in (116) allows to express thetime-lapse in terms of the radar-distance:

∆t∗ = H−1 ln(cos(H l∗/c)

)

≈ (l∗/c){− 1

2 (Hl∗/c) +O(2)}.

(117)

Now, suppose a satellite S moves on a worldline r∗(t∗)in the neighborhood of our worldline r∗ = 0. Assumethat we measure the distance to the satellite by radarcoordinates. Then instead of the value r∗ we would usel∗ and instead of the argument t∗ we would assign thetime t∗ −∆t∗ which corresponds to the value of cosmo-logical time at that event on our worldline that is Einsteinsynchronous to the event (t∗, r∗); see Fig. 2. Hence wehave

l∗(t∗) = (c/H) sin−1{r∗(t∗ +∆t∗)H/c

}(118a)

≈ r∗ − 12 (v/c)(Hc)(r∗/c)

2 , (118b)

where (118b) is (118a) to leading order and all quantitiesare evaluated at t∗. We set v = r∗.To see what this entails we Taylor expand in t∗ around

t∗ = 0 (just a convenient choice):

r∗(t∗) = r0 + v0t∗ +12a0t

2∗ + · · · (119)

and insert in (118b). This leads to

l∗(t∗) = r0 + v0t∗ +12 a0t

2∗ + · · · , (120)

where,

r0 = r0 − (Hc) 12 (v0/c)(r0/c)

2 (121)

v0 = v0 − (Hc) (v0/c)2(r0/c) (122)

a0 = a0 − (Hc){(v0/c)

3 + (r0/c)(v0/c)(a0/c)}

(123)

These are, in quadratic approximation, the sought-afterrelations between the quantities measured via radartracking (tilded) and the quantities which arise in the(improved) Newtonian equations of motion (not tilded).In particular, the last equation (123) shows that there

is an apparent inward pointing acceleration, given by Hctimes the (v/c)3 + · · · term in curly brackets. As dis-cussed in the introduction, Hc is indeed of the same or-der of magnitude as the PA, as was much emphasizedin (Rosales and Sanchez-Gomez, 1998; Rosales, 2002).However, in contrast to these authors, we also gets the

25

A

A′

p0

p1

p2

observer: r∗ = 0

satellite: r∗(t∗)

——— t∗ = const.

∆t∗

FIG. 2 An observer moves on the geodesic worldline r∗ = 0and observes a satellite by exchanging electromagnetic signals(dashed line). Event p0 corresponds to the event’s emissionby the observer, p1 to its reflection by the satellite, and p2 toits re-absorption by the observer. t∗ denotes the cosmologicaltime and the two curved lines correspond to hypersurfaces ofconstant t∗. On the observer’s worldline event A is definedto be simultaneous to p1 with respect to t∗ and A′ is definedto lie half-way in proper time between p0 and p2 on the ob-server’s worldline. In cosmological time A′ is ∆t∗ ahead of A.In cosmological coordinates, the satellite’s trajectory is rep-resented by the t∗ 7→ r∗(t∗), where r∗(t∗) denotes the propergeodesic distance in the hypersurface t∗ = const. betweenits intersection points with both worldlines. However, usingradar coordinates, the observer takes A′ to be simultaneouswith p1 and uses l∗ as measure for the satellite’s simultane-ous distance. Since r∗ and l∗ are related by (116a), it followsthat the observer uses the function t∗ 7→ l∗(r∗(t∗ −∆t∗)) tocharacterize the satellite’s trajectory, which leads to (118).

additional term in curly brackets, which in case of thePioneer spacecraft suppresses the Hc term by 13 ordersof magnitude!17 Hence we conclude that, with respect tothe PA, there is no significant kinematical effect result-ing from the distinct simultaneity structures inherent inradar and cosmological coordinates.

17 Our Eq. (117) corresponds to Eq. (10) of(Rosales and Sanchez-Gomez, 1998). From it the authorsof (Rosales and Sanchez-Gomez, 1998) and (Rosales, 2002)immediately jump to the conclusion that there is “an effectiveresidual acceleration directed toward the center of coordinates;its constant value is Hc”. We were unable to follow thisconclusion. Likewise, we are unable to follow the conclusion in(Fahr and Siewert, 2008).

B. Doppler tracking in cosmological spacetimes

Doppler Tracking is a common method of tracking theposition of vehicles in space. It involves measuring theDoppler shift of an electromagnetic signal sent from aspacecraft to a tracking station on Earth. This signalis either coming from an on-board oscillator or is coher-ently transponded by the vehicle in response to a signalreceived from the ground station. Here we focus on thesecond of these modes, which is more useful for navi-gation, partly because the returning signal is measuredagainst the same frequency reference as that of the orig-inally transmitted signal and partly because the Earth-based frequency reference is also more stable than theoscillator on-board the spacecraft.

1. Minkowski spacetime

It is clear that this method will be fundamentally in-fluenced if performed within a time varying backgroundgeometry. Before elaborating on this, we consider thesimple case of static Minkowski space.

— vehicle

— observer

p0

p1

p2

FIG. 3 Exchange of electromagnetic signals (represented bytheir rays at a slope of 45 degrees) between us and the spacevehicle. Time runs vertically.

In Fig. 3 we depicted two worldlines, one of the ob-server (straight vertically) and one for the space vehicle.A light signal is emitted by the observer at the event p0,reflected by the vehicle at event p1, and finally receivedback by the observer at event p2. We choose a globalMinkowski frame, that is global coordinates {xµ} ={t, xi} with g(∂/∂xµ,∂/∂xν) = diag(1,−1,−1,−1), inwhich the observer (for simplicity assumed to be iner-tial) is at rest at the origin of the spatial coordinates. Ifβ := v/c denotes the radial velocity of the vehicle in unitsof c, the well known special-relativistic Doppler formula

26

(applied twice) says that the ratio between the receivedand the emitted frequencies is18

ω2(t2)

ω0(t0)=

1− β(t1)

1 + β(t1). (124)

Here t0, t1, and t2 refer to the global Minkowski time at-tributed to p0, p1, and p2, respectively. In Doppler track-ing one is interested in the derivative of this ratio withrespect to t2, which yields a measure for the velocity ofthe spacecraft. We assume ω0 to be constant in time andnote that, given the worldlines of the observer and thevehicle, t1 and t0 are uniquely determined by t2 (sincethe events p1 and p0 are determined by p2). If r denotesthe spatial radius coordinate we have t2 − t1 = r(t1)/c.Differentiation with respect to t1 leads to

dt1dt2

=1

1 + β(t1)(125)

and hence

− 1

2

ω2(t2)

ω0(t0)= β(t1)

(1 + β(t1)

)−3

≈ β(t1)(1− 3β(t1) +O(β2)

).

(126)

This shows that −ω2/2ω0, namely (minus one-half) thederivative of the received to emitted frequency ratio withrespect to the proper time of the receiving observer, givesthe spacecraft’s spatial acceleration up to corrections oforder β. Note that in view of note 18 it would be inap-propriate to call these corrections ‘special relativistic’.The final goal of this section is to derive the general-

ization of (126) for a cosmological spacetime. For this weneed two things: First, we need to know what is the gen-eralization of the concepts of spatial velocity and spatialacceleration in an arbitrary spacetime and, second, weneed to know how electromagnetic signals propagate inan arbitrary spacetime. This is taken care of in the nextparagraph.

2. General setting

In order to generalize the notions of spatial velocityand spatial acceleration to arbitrary spacetime one needs

18 Note that in Special Relativity the Doppler Formula does,of course, not distinguish between moving emitter and mov-ing receiver. So (124) is obtained by squaring the frequency

shiftp

(1 − β)/(1 + β), which is picked up once for the ra-tio ωR(tR)/ω1(t1) (receiver moving relative to the Minkowskiframe) and once for ω2(t2)/ω1(t1) (emitter moving relative toMinkowski frame). Incidentally, exactly the same formula wouldresult in non-relativistic physics if the observer is taken to beat rest with respect to the wave-guiding medium (e.g., theether), which distinguishes the two states of relative motion.Indeed, in this case we have ω1(t1)/ω0(t0) = (1 − β(t1)) andω2(t2)/ω1(t1) = 1/(1−β(t1)), whose product is again just (124).

to introduce a fiducial reference ‘observer-field’; compare,e.g., (Bini et al., 1995) and also (Carrera, 2008)). An ob-server at the event p is a future pointing unit timelikevector in the tangent space Tp(M) of M at p. An ob-server field is a field of future pointing unit timelike vec-tors. Any observer u at p gives rise to an orthogonal splitof the tangent space Tp(M) at p in a part parallel to u(the local time-axis) and a part orthogonal to it (the localrest space). Since u is not lightlike the two orthogonalsubspaces are complementary, that is, together they spanthe whole tangent space and intersect only in the zerovector. The orthogonal projections of an arbitrary vec-tor X ∈ Tp(M) onto these subspaces are, respectively,given by

Qu(X) := g(X,u)u , (127a)

Pu(X) := X − g(X,u)u , (127b)

which imply the decomposition identity X = Qu(X) +Pu(X).If two observers u and v are defined at the same point,

the spatial velocity (over c) of v with respect to u is givenby

βu(v) :=Pu(v)

‖Qu(v)‖=

v − g(v,u)u

g(v,u), (128)

which is an element of the local rest space PuT (M). Itsmodulus is given by

βu(v) := ‖βu(v)‖ =√

1− 1/g(u,v)2 . (129)

Note that for the modulus we have βu(v) = βv(u),though the vectors βu(v) and βv(u) are linearly indepen-dent as they lie in PuT (M) and PvT (M), respectively.

Note also that g(u,v) = 1/√

1− β2u(v) is just the ordi-

nary ‘gamma-factor’. Finally, if e ∈ PuT (M) is a unitvector, we define the spatial velocity of v in direction ew.r.t. u by

βeu(v) = −g(e,βu(v)) = − g(e,v)

g(u,v). (130)

The spatial acceleration of a worldline γ w.r.t. a givenobserver field u is defined as the rate of change of the spa-tial velocity βu(γ) within the local rest spaces PuT (M)of u and with respect to the clocks moving along u. De-noting this acceleration (divided by c) with α, we have

αu(γ) := ∇uγβu(γ) , (131)

where we used the following covariant derivative forPuT (M)-valued vector fields along γ:

∇uγ := ‖Quγ‖−1 Pu ◦ ∇γ ◦ Pu . (132)

Here ∇γ denotes the ordinary (Levi-Civita) covariantderivative along γ. As application one can, for example,rewrite the geodesic equation, ∇γγ = 0, for a world-line γ in terms of the spatial quantities just introduced.

27

One gets (see (Carrera, 2008) or, in a slightly differentnotation, (Bini et al., 1995)):

αu = −Sβu

[au + θu(βu) + ωu(βu)

], (133)

where for better readability we omitted the argumentsγ and γ in the spatial acceleration and spatial velocity.Here au := ∇uu is the four-acceleration of the observerfield u, θu and ωu are, respectively its shear-expansionand the rotation tensors of rank (1, 1) (endomorphism),and Sβu

:= Pu+βu⊗βu is a rank (1, 1) tensor which, ina slow-motion approximation (neglecting quadratic andhigher terms in β), reduces to the identity on the lo-cal rest space of u. Equation (133) should be seen asa local version of Newton’s equation. For example, inSchwarzschild–deSitter spacetime (59), taking u to beproportional to the timelike Killing field ∂/∂T , one hasθu = ωu = 0 (because of the Killing equation and spher-ical symmetry, respectively) and

au = ∇uu =1√V

(m

R2− Λ

3R

)

eR , (134)

where eR denotes the the normalized radial vector field∂/∂R (we use here the coordinates and the notationof (59)). Hence, in slow-motion and weak-field approxi-mation (that is keeping only linear terms in β, m/R, andΛR2), the geodesic equation of motion in the form (133)reduces to

αu ≈(

− m

R2+

Λ

3R

)

eR , (135)

which just gives the ‘improved’ Newtonian equation forgeodesic motions in Schwarzschild–deSitter spacetime.It has the same form as the improved Newtonian equa-tion studied in Section III.We turn now to electromagnetic signals and restrict

our attention to monochromatic waves in the geometric-optics approximation (i.e. for wave-lengths negligiblysmall w.r.t. a typical radius of curvature of the space-time and w.r.t. a typical length over which amplitude,polarization, and frequency vary). In this approxima-tion an electromagnetic signal propagates on a light-like geodesic along which the wave-vector, k, is tangent,future-pointing, and parallelly transported. Recall thatk is so normalized that the frequency measured by anobserver, say u, is

ωu(k) := g(u,k) . (136)

Given a wave-vector k and two observers u,v at thesame spacetime point, their observed frequencies are thusωv(k) = g(v,k) and ωu(k) = g(u,k), and their ratio isgiven by

ωv(k)

ωu(k)=

g(Quv + Puv,k)

g(u,k)= g(u,v)

[1− βk

u(v)].

(137)

Here the spacelike unit vector k := ‖Pu(k)‖−1Pu(k) de-fines the direction of k in the local rest space of u. In

deriving (137) we used (130) and ‖Pu(k)‖ = g(u,k)

to write g(v,Pu(k)) = −g(v,u)g(u,k)βku(v). Equa-

tion (137) is the general form of the Doppler formula.Let now u be an observer field along one integral line of

which the distinguished observer is moving. The world-line of the vehicle is denoted by γ. The domain of thefield u is assumed to include a neighborhood of γ. Thewave-vector k0 emitted at p0 suffers three changes:

1. propagation from p0 to p1: k0 → k1;2. reflection at p1: k1 → k′

1;3. propagation from p1 to p2: k

′1 → k2.

We are interested in the ratio of the received to the emit-ted frequency:

ω2

ω0=

g(u2,k2)

g(u0,k0)=

(ω2

ω′1

)(ω′1

ω1

)(ω1

ω0

)

. (138)

What happens at reflection (the second process: k1 →k′1)? Well, with respect to the spacecraft moving along γ

with four-velocity v = γ, the wave vector k1 at p1 splitsaccording to

k1 = Qγ(k1) + Pγ(k1) . (139)

A corner-cube reflector transported along γ will reversePγ(k1) while keeping Qγ(k1) intact (here we neglect apossible transponder shift which is irrelevant for our dis-cussion):

k1 7→ k′1 = Qγ(k1)− Pγ(k1) = 2Qγ(k1)− k1 . (140)

Hence ω1 := ωu(k1) = g(u1,k1) and ω′1 := ωu(k

′1) =

g(u1,k′1), the in- and out-going frequencies measured by

the observer u at p1, are related by

ω′1

ω1= 2

g(u, γ)g(γ,k)|p1

g(u,k)|p1

− 1 = 21− βk

u(γ)|p1

1− β2u(γ)|p1

− 1 ,

(141)where in the last step we just used (137) to rewrite theratio g(γ,k)/g(u,k). This accounts for the middle ratioon the right-hand side of (138).To account for the other two ratios in (138), one uses

the laws of geometric optics in (curved) spacetime to re-late ω0 = g(u0,k0) (at p0) and ω2 = g(u2,k2) (at p2) tokinematical quantities of γ at p1.

19 For example, if u isa Killing field (like u = ∂/∂t in Special Relativity), wehave g(u0,k0) = g(u1,k1) and g(u2,k2) = g(u1,k

′1), so

that

ω2

ω0= 2

1− βku(γ)|p1

1− β2u(γ)|p1

− 1 . (142)

As a trivial application, this includes the generalized formof (124), the latter corresponding to purely radial motion.

19 In general spacetimes without timelike conformal Killing fieldsthese quotients will also explicitly depend on time.

28

3. FLRW spacetimes

In standard cosmological spacetimes (FLRW), u =∂/∂t is not Killing, though X = a(t)∂/∂t is confor-mally Killing (LXg = 2ag). One now has a0g(u0,k0) =a1g(u1,k1) and a2g(u2,k2) = a1g(u1,k

′1), so that in-

stead of (142) one gets

ω2

ω0=

a0a2

{

21− βk

u(γ)|p1

1− β2u(γ)|p1

− 1

}

. (143)

We now want to relate the t2-derivative of (143) tothe acceleration of γ. In order to calculate the derivativeω2(t2)/ω0(t0) we need to know the derivatives dt1/dt2and dt0/dt2. Restricting to the flat FLRW case for sim-plicity, they follow from the law of null propagation:

∫ t2

t1(t2)

dt

a(t)= −1

c

∫ r2

r1(t1(t2))

dr , (144a)

∫ t2

t0(t2)

dt

a(t)=

1

c

{∫ r1(t1(t2))

r0

dr−∫ r2

r1(t1(t2))

dr

}

.

(144b)

Differentiation with respect to t2 yields, respectively

dt1dt2

=a(t1)

a(t2)

(

1 + βku(γ)|p1

)−1

, (145a)

dt0dt2

=a(t0)

a(t2)

1− βku(γ)|p1

1 + βku(γ)|p1

. (145b)

The exact formula for the t2–derivative of thefrequency-shift rate can now be computed. One obtains

− ω2(t2)

ω0(t0)=

a0a2

{

2[αk − g(β,∇u

γ k)]a1a2

[1 + βk

]−1[1− β2

]−1

+ 4g(α,β)a1a2

[

1− βk

1 + βk

]

[1− β2

]−2

+

[

a2a2

− a0a2

(

1− βk

1 + βk

)][

1− 2βk + β2

1− β2

]}

,

(146)

where we suppressed the argument γ and index u atβ for better readability. This formula provides an ex-act relation between the time derivative of the observ-able frequency shift (defined ‘here’) and the kinemati-cal quantities of the vehicle (defined ‘there’), providedthe scale function a(t) is known. For purely radial mo-

tion ∇uγ k = 0 and we obtain the simpler expression (now

writing α for αk)

− 1

2

ω2(t2)

ω0(t0)= − a0a1

a22

{

α(1 + β)−3+

1

2

[a2a1

− a0a1

(1− β

1 + β

)][1− β

1 + β

]}

.

(147)

In order to consistently approximate this expression interms of small quantities β and H∆t, where ∆t :=(t2 − t0)/2, we think of (147) as being multiplied with∆t and regard α∆t as being of order β. Then, keepingonly quadratic terms in β, linear terms in H∆t where∆t := (t2 − t0)/2, and also mixed terms βH∆t, we get

− 1

2

ω2(t2)

ω0(t0)≈ α

(1− 3β − 3H∆t

)+Hβ . (148)

Hence we see that in this approximation there are twomodifications, besides the −3β-term already familiarfrom (126), due to cosmic expansion: First, there is anadditional contribution −3H∆t acting in the same wayas the −3β-term. It can also be interpreted in the samefashion, as its corresponds to the velocity (over c) ofH∆tthat a comoving systems picks up during the time the sig-nal went from the observer to the vehicle. Second, thereis a constant contribution Hβ to acceleration/c, i.e. Hcβto acceleration, in a direction parallel to the radial ve-locity (i.e. outward pointing if the vehicle recedes fromthe observer). Hence it acts opposite to the PA and issmaller in modulus by a factor of β. Applied to the Pi-oneer spacecrafts, the H∆t - term amounts to a tiny‘anomalous’ acceleration of ∆a/a < 10−12, the Hcβ -term to ∆a/a < 10−7.A final point must be made regarding the choice of the

reference observer-field on which the kinematic quantitiesrelated to the spacecraft (spatial velocity and spatial ac-celeration) and the electromagnetic signal (frequency andspatial propagation direction) crucially depend. In theMinkowskian case the reference field was just u = ∂/∂t,which is inertial, that is, geodesic and of vanishing ro-tation, shear, and expansion. It is clear that in a gen-eral spacetime such observer fields do not exist and thereis no natural choice to replace them. However, in thecase of spherical symmetry there is, in fact, a distin-guished observer field, namely that one whose orbits liewithin the timelike hypersurfaces of constant areal ra-dius and there run perpendicular to the orbits of therotation group. This clearly defines a non-rotating and‘non-expanding’ (w.r.t. the areal radius) reference field.It is the normalization of the so-called Kodama vectorfield, which we discuss in detail in Appendix D.3. In aFLRW spacetime it is just given by (113). Notice thatin the present case, where the hypersurfaces of constantcosmological time t are flat, the areal radius correspondsalso to the proper distance. Hence the integral curvesof u∗ intersect the hypersurface of constant cosmologicaltime at constant spatial geodesic distance. More pre-cisely, the expansion and the shear scalar of u∗ are givenby θ∗ = R2HH/(1 − (RH)2)3/2 and σ∗ = −θ∗/3, re-spectively, showing that they are of order H3 which weneglect. In passing we remark that the expansion andshear of u∗ exactly vanish for the de Sitter case, whosemetric in ‘static’ coordinates is given by (59) for m = 0.In this case ∂/∂t∗ = ∂/∂T , that is, u∗ is proportional tothe timelike Killing vector field ∂/∂T ; see (59). Comingback to the general FLRW case, the acceleration of u∗

29

is given by au∗= (−Ra/a + R3H4)/(1 − (RH)2)3/2e∗,

where e∗ is the unit vector field orthogonal to u∗ andto the two-sphere, pointing in positive radial direction.Hence, in the slow-motion and weak-field approximationof Eq. (148), but keeping also quadratic terms in H , thegeodesic equation in the form (133) w.r.t. the observerfield u∗ reads as

αu∗(γ) ≈

(a

ar∗

)

e∗ ◦ γ . (149)

This is just an alternative derivation of the accelerationterm (7). We point out that had we we taken (110) asobserver field we would have arrived at the equation ofmotion αu(γ) ≈ −Hβu(γ) instead of (149), that is, noacceleration term (7) would have resulted.In the approximation within which (148) is derived

this equation remains valid if the quantities in it are re-interpreted so as to refer to u∗ instead of u. Hencewe may sum up the situation by saying that equa-tions (148) and (149) give, respectively, the two-wayDoppler-tracking formula and the ‘Newtonian’ equationin a FLRW spacetime within the mentioned approxima-tion.

4. McVittie spacetime

The same analysis can be generalized from the spatiallyflat FLRW spacetime (30) to the spatially flat McVittiespacetime (68). Here the observer moves along ∂/∂t,which is not geodesic. The coordinate t does now notmeasure proper time, denoted by τ , along the observer’sworldline. The result corresponding to (148) can now bestated as follows:

− 1

2

ω2(τ2)

ω0(τ0)≈ α

(1− 3β − 3∆τ(H −m0c/R

2))+Hβ .

(150)Here R denotes the areal radius of the observer during themeasurement. Note that even though it changes alongthe observer’s worldline according to (90), we do not needto account for the corresponding change in ∆τm0c/R

2

of (−2∆τm0c/R2)(H∆τ) which is of subleading order.

The additional term in (150) has a straightforward inter-pretation in terms of the acceleration that the observernecessarily experiences while keeping a constant radiusR away from the central inhomogeneity.As for the FLRW case, we chose the observer field to

which we refer the spatial quantities to be proportionalto the Kodama vector field (along which the areal ra-dius is constant). Putting r∗(t, r) := A2(t, r)a(t)r andt∗(t, r) := t, a short computation shows that the vectorfield ∂/∂t∗ is again given by (112). In the slow-motionand weak-field approximation used in Section V.D.2, thegeodesic equation in the form (133) w.r.t. the observerfield u∗ reads

αu∗(γ) ≈

(a

ar∗ −

m0

r2∗

)

e∗ ◦ γ , (151)

where again e∗ denotes the unit outward-pointing vec-tor field orthogonal to u∗ and to the two-spheres ofsymmetry. This is an alternative derivation of the im-proved Newtonian equation for the McVittie spacetimecarried out in Section V.D.2. Notice that, again, withinthe approximations used, relation (150) remains valid ifone refers the quantities to u∗ instead of to u. There-fore (150) and (151) give the two-way Doppler-trackingformula and the improved ‘Newtonian’ equation for theMcVittie spacetime within the mentioned approximation.In the special case of purely radial motion, insertion

of (151) into (150) leads to a formula predicting the two-way Doppler-shift rate in linear order inH∆τ andm0/r∗,and quadratic order in βu∗

(γ):

− 1

2

ω2(τ2)

ω0(τ0)= − m0

r2∗

(

1− 3βk)

+Hβk . (152)

Hence there are two corrections to the Newtonian con-tribution. One is proportional to H and stems from thecosmological expansion, the other, already familiar fromthe special-relativistic treatment (126), is independent ofH and merely due to the finiteness of the propagationspeed of light (recall note 18). Their ratio is (up to a

factor√3) given by the square of the ratio of r∗ to the

geometric mean of the Schwarzschild radius m0 and theHubble radius c/H . The latter is of the order of 1023 km,so that its geometric mean with a Schwarzschild radiusof one kilometer is approximately given by 2400 astro-nomical units. The ratio of the effects is therefore of theorder 10−7. Hence the cosmological contribution is negli-gible for any application in the Solar System as comparedto the 3β–correction. For the Pioneer 10 & 11 space-crafts we have a radial velocity of about 12 Km/s. Thisamounts to a 3β–correction of magnitude 4 · 10−5 timesthe Newtonian gravitational acceleration, in an outward-pointing direction. This is indeed of the same order ofmagnitude as the PA but directed oppositely.

VII. SUMMARY AND OUTLOOK

We think it is fair to say that there are no theoret-ical hints that point towards a dynamical influence ofcosmological expansion comparable in size to, say, thatof the anomalous acceleration of the Pioneer spacecrafts.There seems to be no controversy over this point, thoughfor completeness it should be mentioned that there ex-ist speculations (Palle, 2005) according to which it mightbecome relevant for future missions. But such specu-lations are often based on models which are not eas-ily related to the intended physical situation, like thatof Gautreau (Gautreau, 1984). Rather, as the (a/a)–improved Newtonian analysis in Section III together withits justification given in the subsequent Sections shows,there is no genuine relativistic effect coming from cos-mological expansion at the levels of precision envisagedhere.

30

On the other hand, as regards kinematical effects, thesituation is less unanimous. It is very important to un-ambiguously understand what is meant by ‘mapping outa trajectory’, i.e. how to assign ‘times’ and ‘distances’.Eventually we compare a functional relation between‘distance’ and ‘time’ with observed data. That relation isobtained by solving some equations of motion and it hasto be carefully checked whether the methods by which thetracking data are obtained match the interpretation ofthe coordinates in which the analytical problem is solved.In our way of speaking, dynamical effects really influencethe worldline of the object in question whereas kinemat-ical effects change the way in which one and the sameworldline is mapped out from another worldline repre-senting the observer. Here we have derived exact resultsconcerning the influence of cosmic expansion on this map-ping procedure, which allow to reliably estimate upperbounds on their magnitude. They turn out to be toosmall to be of any relevance in current satellite track-ings, which is an accord with naive expectation but incontrast to some statements found in the literature.

At this point it is useful to recall the general philoso-phy behind such statements: From the Einstein–Straussolution it is clear that local overdensities inhibit cosmicexpansion. For example, calculating its effect in simplemodels like the improved Newtonian equation discussedin Section II.A (backed up by the various justificationswe discussed in detail) clearly means to overestimate theimpact of cosmic expansion in a realistic situation, wherethe single overdensity (e.g. representing the Sun) is sur-rounded by more overdense structures (the Solar-Systemenvironment, the Galaxy, etc.). If this overestimationgives an already insignificant upper bound for the envis-aged effect, we can conclude that it becomes even moreinsignificant in more realistic models.

Satellite navigation is clearly not the only potentialsource of interest in the question of how local inhomo-geneities affect cosmological expansion. Many predic-tions concerning cosmological data rely on computationswithin the framework of the standard homogeneous andisotropic models, without properly estimating the possi-ble effects of local inhomogeneities. Such an estimationwould ideally be based on an exact inhomogeneous solu-tion to Einstein’s equations, or at least a fully controlledapproximation to such a solution. The dynamical andkinematical impact of local inhomogeneities might essen-tially influence our interpretation of cosmological obser-vations. As an example we mention recent serious effortsto interpret the same data that are usually taken to provethe existence of a positive cosmological constant Λ ina context with realistic inhomogeneities (Buchert, 2000;Wiltshire, 2007), i.e. taking into account that cosmologi-cal parameters are dressed (Buchert and Carfora, 2003).One might speculate that the measured Λ can eventu-ally fully reduced to the action of inhomogeneities, assuggested in in (Wiltshire, 2007, 2008). For an earlieradvance in this direction, see (Celerier, 2000).

Acknowledgments

This work was partially supported by the EuropeanSpace Agency (ESA) under the Ariadna scheme of theAdvanced Concepts Team, contract 18913/05/NL/MV.We are grateful to the ESA and the Albert-Einstein-Institute in Golm for their support and hospitality.

APPENDIX A: Notation, conventions, and generalities

Amodel for spacetime consists of a tuple (M, g), whereM is a four-dimensional manifold and g a Lorentzianmetric whose signature we take to be (+,−,−,−), i.e. weuse the ‘mostly minus’ convention. Throughout we de-note geometric objects, like tensor fields and covari-ant derivative operators, by bold-faced letters or words.The unique metric preserving and torsion-free covariantderivative associated with g will be denoted by ∇ andthe covariant derivative in the direction of a vector Xby ∇X . For a smooth tensor field T on M its covariantderivative ∇T defines a linear map, X 7→ ∇XT , fromthe tangent space to the tensor space at each point ofM where T is defined. Since ∇T is again a tensor field(of rank (p, q + 1) if the rank of T was (p, q)) we canform ∇∇T := ∇(∇T ). Note that (∇∇T )(X,Y ) =∇X∇Y T − ∇∇XY T . For a scalar function f on Mwe have ∇f = df , the ordinary exterior differential, and∇∇f = Hess(f), the Hessian of f . The metric g al-lows to uniquely associate to any vector X a linear formX := g(X, · ), called the dual (with respect to g) of X.Metricity of ∇ then implies ∇XY = ∇XY .Associated with any two linearly independent vectors

X,Y at a point p ∈ M is a curvature endomorphism,R(X,Y ), of the tangent space at p:

R(X,Y )Z = (∇∇Z)(X,Y )− (∇∇Z)(Y ,X)

= ∇X∇Y Z −∇Y ∇XZ −∇[X,Y ]Z .(A1)

The Riemann- or curvature tensor, Riem, is then definedby

Riem(W ,Z,X,Y ) := g(W ,R(X,Y )Z

). (A2)

It is antisymmetric under the exchange X ↔ Y or W ↔Z and symmetric under the slotwise exchange of pairs(W ,Z) ↔ (X,Y ). Moreover, the antisymmetrizationover any three slots vanishes (first Bianchi identity). TheRicci tensor, Ric, is defined by the trace of the followingendomorphism

Ric(Y ,Z) := tr(X 7→ Riem(X,Y )Z

), (A3)

which is symmetric under exchange Y ↔ Z. The scalarcurvature is defined by taking the trace ofRic, also calledthe Ricci scalar, with respect to g (since Ric is not anendomorphism, we need the metric to define its trace)

Scal = trg(Ric) . (A4)

31

Finally, the Einstein tensor is the following combinationof Ric and Scal:

Ein := Ric− 12 Scal g . (A5)

Associated to any spacelike or timelike two-dimensional plane Π in the tangent space at p ∈ M isthe sectional curvature. Its geometric interpretation isjust that of the ordinary Gaussian curvature at p of thetwo-dimensional surface in M that is spanned by thegeodesic curves through p tangent to Π . In terms ofRiem it reads

kΠ :=Riem(X,Y ,X,Y )

Q(X,Y ), (A6)

where X,Y are any two linear independent vectors in Πand

Q(X,Y ) : = g(X,X)g(Y ,Y )− g(X,Y )2

= (g ⊙ g)(X,Y ,X,Y ) .(A7)

Note that |Q(X,Y )| gives the square of the area of theparallelogram spanned by X and Y which is nonzeroiff the considered plane is spacelike or timelike (non-degenerate).In (A7) we introduced the product ⊙, which is called

the Kulkarni–Nomizu product. It is a symmetric bilinearmap from the space of symmetric (0, 2) tensors to thespace of (0, 4) tensors with the same algebraic symmetriesas Riem. Its general definition is as follows:

(a⊙ b)(W ,Z,X,Y ) :=

1

2

(

a(W ,X)b(Z,Y )− a(W ,Y )b(Z,X)

+b(W ,X)a(Z,Y )− b(W ,Y )a(Z,X))

.

(A8)

This can be used to conveniently write down the g-orthogonal decomposition of the curvature tensor intothe Ricci- and the Weyl part:

Riem = Ricci+Weyl . (A9)

In four spacetime dimensions one has

Ricci : =(Ric− 1

6 Scal g)⊙ g (A10a)

=(Ein− 1

3 trg(Ein) g)⊙ g . (A10b)

Inserting this into (A9) gives the definition ofWeyl. Thedefinition is such that the Ricci part is g-orthogonal tothe Weyl part and that the latter is totally trace free.Hence the Ricci and the Weyl part each contribute 10independent components to the 20 independent compo-nents of Riem. The Ricci part may be further decom-posed according to the decomposition of Ric into itstrace and a trace-free part, but this refinement will notbe needed here.Einstein’s equation now express the local determina-

tion of the Ricci part of the curvature in terms of the

energy-momentum distribution of matter, the latter be-ing encoded in the energy-momentum tensor T of thematter. In units where Newton’s constant G and thevelocity of light c equal one20, Einstein’s equation reads

Ein = 8π T . (A11)

Here we did not write down explicitly a cosmologicalterm, which can always be thought of as extra contri-bution to T of the form g Λ/8π. Now, assuming that gsatisfies Einstein’s equation, the Ricci part of the Rie-mann tensor is given in terms of T by

Ricci = 8π(T − 1

3 trg(T ) g)⊙ g . (A12)

APPENDIX B: Proof of Theorem 1

In this section we prove Theorem 1, namely the equiv-alence, in the spherically-symmetric case, of the SSJCwith the Darmois junction conditions.

Proof. The proof essentially consists in writing down theinduced metric and extrinsic curvature for a (non-null)spherically symmetric hypersurface in a spherically sym-metric spacetime. This is most easily done by introducingan adapted orthonormal frame.We first consider the case where Γ is timelike, hence

γ = π(Γ ) is a timelike curve in B. The following con-struction shall be carried out in both spacetimes. Onedefines v as in the SSJC, hence as the (unique up to asign) spherically symmetric, unit vector field on Γ or-thogonal to n. That is v, seen as a vector field on B,is tangent to γ. Since n is spacelike, v is timelike. Theambient metric can be then written as

g = v ⊗ v − n⊗ n−R2gS2 , (B1)

so that the induced metric (compare Appendix C and(C3) on Γ is

gΓ = v ⊗ v −R2gS2 . (B2)

In view of (C3) note that here ε(n) = −1. For the ex-trinsic curvature (C6), using (D3) and the fact that v isspherically symmetric and hence tangent to B, one hasthe decomposition

KΓ = −g(n,∇vv)v ⊗ v −RdR(n) gS2 . (B3)

Now, from expressions (B2) and (B3) it follows thatthe DJC, and hence the continuity of gΓ and KΓ , areequivalent to the continuity of the following four func-tions:

(a) the arc-length of γ,

20 Otherwise the factor 8π on the right-hand side of (A11) shouldbe replaced with 8πG/c4.

32

(b) R,(c) dR(n), and(d) g(n,∇vv).

The statement of the theorem will now follow from thefollowing expression of the MS energy (58):

E =R

2

(1 + (dR(v))2 − (dR(n))2

). (B4)

Simply note that if R is continuous through Γ (recallthe definition of this concept below the definition of DJCin Section V.A) the same holds for its derivatives tan-gent to Γ . In particular, dR(v) is continuous through Γand hence we may substitute dR(n) by the MS energyin the above list (a)–(d). This completes the proof fortimelike Γ .In the case of spacelike Γ the unit normal n is timelike

and v is chosen as the unique (up to a sign) sphericallysymmetric unit vector field on Γ orthonormal to n. Thenv is a spacelike ‘radial’ unit vector field orthogonal tothe SO(3) orbits. The proof now proceeds analogouslyto the timelike case. We merely list the expressions forthe ambient metric

g = n⊗ n− v ⊗ v −R2gS2 ,

the induced metric

gΓ = v ⊗ v +R2gS2 , (B5)

the extrinsic curvature

KΓ = g(n,∇vv)v ⊗ v +RdR(n) gS2 , (B6)

and the MS energy

E =R

2

(1 + (dR(n))2 − (dR(v))2

), (B7)

and conclude exactly as in the timelike case.

APPENDIX C: Submanifolds

In a Lorentzian manifold (M, g) endowed with Levi-Civita connection ∇ consider a smooth submanifold Γof codimension one and normal vector field n. We as-sume Γ to be non-null, that is, either spacelike (then nis timelike) or timelike (then n spacelike). Then Γ in-herits from the ambient manifold M a (non-degenerate)metric and a connection in a natural way. We introducethe orthogonal projectors

Qn := ε(n)n⊗ n (C1a)

Pn := id−Qn , (C1b)

where ε(n) denotes the indicator, defined for any non-null vector by

ε(X) :=g(X,X)

|g(X,X)| ={

+1 if X is timelike,

−1 if X is spacelike.(C2)

The induced metric on Γ (also called first fundamentalform) is given by

gΓ := −ε(n)Png , (C3)

where the sign is just in order to get a positive definitemetric in the case where Γ is spacelike. Given two vectorfields X,Y tangent to Γ , so that QnX = QnY = 0,one may decompose the covariant derivative of Y withrespect to X into its orthogonal components

∇XY = Pn(∇XY ) +Qn(∇XY )

= Γ∇X Y +KΓ (X,Y )n , (C4)

where

Γ∇X Y := Pn(∇XY ) (C5)

is the induced connection on Γ and

KΓ (X,Y ) := ε(n) g(∇XY ,n)

= −ε(n) g(∇Xn,Y ) (C6)

is the extrinsic curvature of Γ in M (also called the sec-ond fundamental form). The second equality sign in (C6)is an immediate consequence of the metricity of ∇ andthe fact that X and Y are orthogonal to n. With thisalternative expression for the extrinsic curvature one hasKΓ = −ε(n)Pn∇n and hence21

KΓ = −ε(n)S(Pn∇n) . (C7)

Since n is hypersurface orthogonal (by definition) wehave A

(Pn∇n

)= 0. Hence, the extrinsic curvature is a

symmetric (0, 2)-tensor field. We recall also that the in-duced connection is the Levi-Civita connection of (Γ, gΓ ),as one may easily check.

The full relations between the curvature of M andthose (intrinsic and extrinsic) of Γ can be found,e.g., in (Giulini, 1998). Here we are only interested inthe ‘Einstein part’ of the curvature. One gets:

Ein(n,n) =1

2

(

−ε(n) ΓScal+(trK)2 − ‖K‖2)

(C8a)

Ein(n,Pn· ) = −ε(n) divΓ

(K − (trK) gΓ

)(C8b)

Ein(Pn· ,Pn· ) = ΓEin

+ ε(n)

(1

2

((trK)2 + ‖K‖2

)gΓ − (trK)K

+ Ln

(K − (trK) gΓ

))

. (C8c)

21 Here and below S and A denote the projection operators of fullsymmetrization and full antisymmetrization, respectively.

33

APPENDIX D: Spherical symmetry

We recall that the isometry group, Isom(M, g), of aspacetime (M, g) is the subgroup of the diffeomorphismgroup of M, Diff(M), which leaves the metric g invari-ant: Isom(M, g) := {φ ∈ Diff(M) |φ∗g = g}.

Definition 3 (Spherical symmetry). A four-dimensionalLorentzian manifold (M, g) is said to be spherically sym-metric if its isometry group, Isom(M, g), contains a sub-group G with the following two properties: (i) G is iso-morphic to SO(3) and (ii) each orbit of G is spacelikeand two-dimensional (up to some closed proper subset offixed points). A tensor field T on a spherically symmet-ric spacetime is said to be spherically symmetric if it isinvariant under G, hence if φ∗T = T for all φ ∈ G.

From this definition it follows (excluding the casewhere the orbits of G are diffeomorphic to thetwo-dimensional real-projective space) that a four-dimensional spherically symmetric Lorentzian manifold(M, g) can, at least locally, be expressed as a warpedproduct M = B ×R S2 between a two-dimensionalLorentzian manifold (B, gB), called the ‘base’, and thestandard unit two-sphere (S2, gS2), called the ‘fiber’(see (Straumann, 2004) and (O’Neill, 1983)). This meansthat, at least locally, the manifold is a product

M loc= B × S2 (D1)

and the metric is given by

g = π∗(gB)− (R ◦ π)2σ∗(gS2) . (D2)

Here, π and σ are the projections of B×S2 onto B and S2,respectively, and π∗, σ∗ their pull-backs. The warpingfunction R is nothing but the areal radius, since, for apoint p ∈ B, the area of the fiber p× S2 is just 4πR(p)2.In this contest, a vector field X on M at some point

(p, q) ∈ B × S2 has then a unique decomposition X =tanB X+tanS2 X in a component tangent to the ‘leaves’B × q = σ−1(q) and a component tangent to the ‘fibers’p× S2 = π−1(p). Arbitrary tensor fields on B and on S2

can be lifted to tensor fields on M in the standard way.For covariant tensor fields (and hence, in particular, forfunctions) this is achieved via the pull-pack of the respec-tive projection: as an example, just look at (D2). Forcontravariant tensor fields it suffices to consider the spe-cial case of vector fields. Let, for instance, X be a vectorin the tangent space of B at p. Then the lift X at (p, q)of X is defined as the unique vector in the tangent spaceof M at (p, q) with π∗(X) = X and σ∗(X) = 0. Sincethis assignment is smooth, one gets the lifting of a vec-tor field via the pointwise lifting just described. In thiswork, we will mainly omit lifts and projections and notexplicitly distinguish between original and lifted quanti-ties. For example, when referring to a vector ‘tangent toB’ we refer to a vector in the tangent space of B or tothe lift thereof in the tangent space of M.

IfX is spherically symmetric, then the component tan-gent to the fibers must vanish: tanS2 X = 0. Similarly, aspherically symmetric one-form θ on M must necessar-ily be tangent to B (i.e. normal to S2) and thus it canbe written as θ = π∗(θB), where θB is a one-form on B.Finally, a spherical symmetric function is simply the liftof a function on B.

1. Connection and curvature decomposition

In the following we discuss relations which express thecurvature of M in terms of the warping function R andthe curvatures of the base B and the fiber S2. We startout from the relations between the Levi-Civita connec-tion ∇ of (M, g) and the Levi-Civita connections of the

base and the fiber, denoted by B∇ and S2

∇, respec-tively. These relations can be derived, for example, bymeans of the Koszul formula (see e.g. (O’Neill, 1983)Proposition 7.35). Let in the following X,Y ,Z be vec-tor fields tangent to B and U ,V ,W tangent to S2. Sup-pressing lifts and projections, we have:

∇X Y = B∇X Y (D3a)

∇X V = ∇V X = R−1X(R)V (D3b)

tanS2 ∇V W = S2

∇V W (D3c)

tanB ∇V W = −g(V ,W )R−1∇R . (D3d)

Note that, for a function f on B, the lift of the gradi-ent is equal to the gradient of the lifted function, thatis (suppressing the lifts): ∇f = B

∇f . For brevity, wewrite just ∇f for it. Take care that for the Hessianand the Laplacian this is in general not true (see (D13)and (D14)). Therefore we write explicitly the super-scripts ‘B’ in BHess f and B∆ f to denote the Hessianand Laplacian of f on B, respectively, or the lifts thereof.By means of (D3) one can now compute the expres-

sions for the Riemann tensor. As the sectional curvatureof S2 is obviously constant and equal to one, the Rie-mann tensor, the Ricci tensor, and the Ricci scalar of S2

are simply given by S2

Riem = gS2⊙ gS2 , S2

Ric = gS2 ,

and S2

Scal = 2, respectively. Here we made again useof the Kulkarni–Nomizu product (A8). Moreover, sincethe basis manifold B is two-dimensional, one can expressits curvature tensors in terms of the scalar curvature.The expression for the Riemann tensor, Ricci tensor, andRicci scalar of a spherically symmetric Lorentzian mani-fold (D1,D2) are, respectively,

Riem =BScal

2gB⊙ gB −R2

(

1+〈dR,dR〉)

gS2⊙ gS2

+ 2R gS2⊙BHessR , (D4)

Ric =BScal

2gB − 2

RBHessR

+(

1+〈dR,dR〉+R B∆R)

gS2 , (D5)

34

and

Scal = BScal− 2

R2

(

1+〈dR,dR〉)

− 4

RB∆R . (D6)

Hence, for the Einstein tensor we have the expression

Ein =

(1

R2

(1+〈dR,dR〉

)+

2

RB∆R

)

gB − 2

RBHessR

+

(BScal

2− 1

RB∆R

)

R2gS2 (D7)

and for the Weyl tensor, using (A9) and (A10a), the sim-ple expression

Weyl = w (gB ⊙ gB + gB ⊙R2gS2 +R2gS2 ⊙R2gS2) .(D8a)

Here we put

w :=1

6

(

BScal− 2

R2

(1+〈dR,dR〉

)+

2

RB∆R

)

(D8b)

=1

6Scal+

1

RB∆R . (D8c)

In the derivation of (D8) we made use of the formulahB⊙gB = (1/2)(trgB

hB)gB⊙gB, valid22 for any symmet-

ric bilinear form hB on B, in order to express the onlyterm involving BHessR in terms of the Laplacian of R.Note that from (D8a) it is immediate that the Weyl ten-sor of a spherically symmetric spacetime has only oneindependent component, as it must be the case due to itbeing of Petrov-type D.Comparing expression (D4) with the definition of sec-

tional curvature (A6) one can immediately read off thatthe sectional curvature of the plane tangential to the(two-dimensional) SO(3)-orbits at a given point is

k = Riem |S2S2 = − 1

R2

(1 + 〈dR,dR〉

), (D9)

and hence the MS energy, defined as (minus one-half) ktimes the third power of the areal radius, is given by (58).Using the MS energy (58), we can write the Einsteintensor (D7) as:

Ein =2

R

(

B∆R+E

R2

)

gB − 2

RBHessR

+

(BScal

2− 1

RB∆R

)

R2gS2 . (D10)

We conclude giving the decomposition for the diver-gence of a spherically symmetric vector field (that is a

22 To prove this just note that the only independent component ofthis formula is the (e0,e1,e0,e1) one, where {eµ} is an adaptedorthonormal basis of (M, g) such that e0,e1 are tangent to B ande2,e3 are tangent to S2. Then, the equality follows immediatelyusing the definition (A8) of the Kulkarni–Nomizu product.

vector field X tangent to B) and for the Hessian andLaplacian of a spherically symmetric function (that is afunction f on B). First, from (D3a) and (D3b), we obtainthe following decomposition for the covariant derivativeof X (expressed as a (0, 2)-tensor):

∇X = B∇X −X(R)R gS2 . (D11)

Note that the mixed term (B-S2) vanishes—as it shoulddue to spherical symmetry. Taking the trace of (D11)one obtains the following expression for the divergence:

divX = divB X +2

RX(R) =

1

R2divB(R

2X) . (D12)

The decompositions for the Hessian and Laplacian of afunction f on B follow from inserting X = ∇f in theabove formulae. One gets:

Hess f = BHess f − gB(∇f,∇R)R gS2 (D13)

and

∆ f = B∆ f + 2gB(∇f,∇R)/R , (D14)

respectively.

2. Einstein equation in case of spherical symmetry

A general spherically symmetric matter energy-momentum tensor has the form

T = TB + pR2gS2 , (D15)

where p is the spherical part of the pressure. Hence, usingthe decomposition (D10) of the Einstein’s tensor foundin Appendix D, the Einstein equation takes the form

2

R

(E

R2+ B∆R

)

gB − 2

RBHessR = 8πTB (D16a)

BScal

2− 1

RB∆R = 8πp . (D16b)

Using the trace of the first equation,

1

R

(

B∆R+2E

R2

)

= 4π trTB , (D17)

to eliminate B∆R, one can write (D16) in the equivalentform

1

R

(E

R2gB + BHessR

)

= −4π ⋆ TB⋆ (D18a)

BScal

2+

2E

R2= 4π

(trTB + 2p) . (D18b)

Here, and in the following, ⋆ denotes the Hodge-dualitymap for (B, gB) (for the definition, see e.g. (Straumann,2004)). In the first equation we used the identity ⋆τ⋆ =τ − tr(τ ) g, which is valid for any bilinear form τ on B,where the first (second) star acts on the first (second)slot of τ .Finally, the integrability condition divT = 0 for the

energy-momentum tensor (D15) reads

divB(R2TB) + pd(R2) = 0 . (D19)

35

3. Misner–Sharp energy

Let us now turn to the MS energy and its properties.We first show that it is the charge of a conserved current.The treatment presented here follows mainly (Hayward,1996). In a spherically symmetric spacetime one definesthe Kodama vector field (Kodama, 1980) as the (uniqueup to a sign) spherically symmetric vector field orthogo-nal to, and of the same norm as, the gradient of R; hencewe put

k := ⋆dR . (D20)

With this sign choice k is future-pointing if the gradientof R is spacelike. The orthogonality between k and thegradient of R is simply expressed by

k(R) = 0 , (D21)

which clearly means that the integral curves of k stay atconstant areal radius.An immediate but important property of the Kodama

vector field is that it is conserved:

divk = 0 . (D22)

Indeed, using (D12) and (D21), one has: div k =divB k = δk = − ⋆ d ⋆ ⋆dR ≡ 0.Now, a key point for the study of spherically symmet-

ric spacetimes is the following equation relating the MSenergy with the matter’s energy-momentum tensor:

dE = 4πR2 ⋆ j , (D23)

where j is the so-called Kodama current (tangent to thebase manifold B) defined by

j := T (k, ·) . (D24)

Equation (D23) follows from Einstein’s equation; moreprecisely, it is equivalent to its B-part (that is Eq. (D18a))fed with ∇R. To see this, just compute the differentialof (58) as follows: dE = (E/R)dR+ (R/2)d〈dR,dR〉 =(E/R)dR+R BHessR ·dR = R

((E/R2)gB+

BHessR)·

dR, where the dot denotes here the contraction of thelast slot of the tensor on the left with the first slotof the tensor on the right of the dot. In the secondstep we use that d(〈dR,dR〉)(X) = X(g(∇R,∇R)) =2 gB(∇X∇R,∇R) = 2 BHessR(∇R,X) for anyX tan-gent to B. Then, inserting (D18a) and using that ⋆is skew-adjoint on one-forms, one gets dE = −4πR2 ⋆TB ⋆ ·dR = 4πR2 ⋆ TB · ⋆dR and hence, using the defi-nitions (D20) and (D24) together with the symmetry ofT , one arrives at (D23).From (D23) it is clear that

j(E) = 0 , (D25)

which means that the vector field j is tangent to thecurves in B (hypersurfaces in M) of constant MS energy.Moreover, (D23) implies that j is also conserved

div j = 0 , (D26)

where the divergence is here taken on the spacetime(M, g). To see this, just compute the divergence of jwith (D12) and using the Hodge-dual version of (D23):div j = R−2 divB(R

2j) = R−2δ(R2j) = (4πR2)−1δ ⋆dE ≡ 0.Following (Hayward, 1996), we can now show that the

charges corresponding to the conserved currents j andk are, respectively, the MS energy and the areal vol-ume. Let Σ be some spatial three-dimensional hypersur-face which, because of spherical symmetry, decomposesas Σ = σ×S2, where σ is some spatial curve in B. Recallthat the charge related to a conserved current X is givenby QX(Σ) :=

∫

Σ iXµ, where µ is the volume form onM, and, because of spherical symmetry, the latter de-composes as µ = µB∧R2µS2 , where µB and µS2 are thevolume forms on B and on the unit two-sphere, respec-tively. After integration of the spherical part and sinceijµB = ⋆j, using (D23) one gets

Qj(Σ) =

∫

σ

dE , (D27)

which means that the charge of j is the MS energy. Thisjustifies the interpretation of the MS energy as a quantityassociated to the ‘interior’ of the considered sphere ofsymmetry. In fact, due to (D26), the charge does notdepend how one choose the spatial slice to define theinterior. Similarly, since ikµB = ⋆k = dR, the charge tok is simply

∫

σ4πR2dR and hence

Qk(Σ) =

∫

σ

d

(4π

3R3

)

, (D28)

which says that the charge of k is the flat-space volumecomputed with the areal radius.Incidentally, the Kodama vector can be used to give

an elegant proof of Birkhoff’s theorem, which states thatspherically symmetric solutions of Einstein’s equationsare, in fact, static. Indeed, by direct computation oneshows that in vacuum k is Killing and, because of spher-ical symmetry, it is clearly also hypersurface orthogonal.Next we turn to the relation between the MS energy

and the Hawking quasi-local mass (Hawking, 1968). Thelatter is a quantity associated to a spatial two-sphere, S,in an arbitrary spacetime. It is defined by

MH(S) :=

√

Area(S)

16π

(

1 +1

2π

∫

S

θ+θ−µS

)

. (D29)

Here, θ± := trS2(∇l±)/2 are, respectively, the expan-sions of the outgoing and ingoing future-pointing nullvector fields l± normal to S, the latter being partiallynormalized such that g(l+, l−) = 1 (there remains thefreedom to rescale l± → α±1l±, where α is a positivereal number). In the special case of spherical symmetrywe take S to be an orbit of the rotation group. Thenwe clearly have Area(S) = 4πR2. It is also obvious thatthe metric of the base B, evaluated on S, can simply bewritten in the form

gB = l+ ⊗ l− + l− ⊗ l+ . (D30)

36

Now, for V tangent to S, (D3b) gives ∇V l± =R−1l±(R)V so that θ± = R−1l±(R). Hence we have:

2 θ+θ− = 2R−2 dR(l+)dR(l−) = 〈dR,dR〉/R2 , (D31)

where we used (D30), or rather its contravariant version,in the last step. Equation (58) now establishes the equal-ity between the MS energy at p and the Hawking quasi-local mass of S, where p is any point on S:

E(p) = MH(S) . (D32)

As is the case for the Hawking quasi-local mass, theMS energy can be naturally decomposed into a Ricci anda Weyl part:

E = ER + EW , (D33)

where

ER := − 12R

3Ricci |S2S2 , (D34a)

EW := − 12R

3Weyl |S2S2 . (D34b)

The Ricci part is determined by the local matter distri-bution via Einstein’s equation: Using expressions (D15)and (D2) for an arbitrary spherically symmetric energy-momentum tensor and, respectively, metric in (A12) onegets

ER =4π

3R3(trTB + p) . (D35)

For the Weyl part of the MS energy we have, in viewof (D8a), that

EW = − 12R

3w , (D36)

where w is given by (D8b) or (D8c). Hence, in particular,the Weyl tensor vanishes iff EW does. Since the squareof the Weyl tensor is 〈Weyl,Weyl〉 ≡ WαβγδW

αβγδ =12w2, with (D36) we obtain the nice expression

〈Weyl,Weyl〉 = 48E2

W

R6. (D37)

4. Spherically symmetric perfect fluids

We specialize now to a perfect fluid, which is describedby a four-velocity vector field u, density , and pressurep. In case of spherical symmetry u is tangent to the basismanifold and the matter energy-momentum tensor (70)decomposes as

T = u⊗ u+ p (u⊗ u− gB) + pR2gS2 , (D38)

from which one can read off the part tangent to B:

TB = u⊗ u+ p (u⊗ u− gB) . (D39)

Usually, the description is to be completed with the spec-ification of an equation of state. We will not assume any

equation of state yet, since in some cases (e.g. McVittiespacetime) this happens to be determined by Einstein’sequation.Inserting (D39) in (D35) we get for the Ricci part of

the MS energy the simple expression

ER =4π

3R3 . (D40)

Also the expression (D23) for the differential of the MSenergy simplifies in case of a perfect fluid. Using (D39)the Kodama current (as one-form) becomes

j = (+ p)g(k,u)u− pk . (D41)

It is useful to introduce an adapted orthonormal basis{u, e} tangent to the basis manifold, where u is the ve-locity vector field of the fluid and e is chosen to pointin direction of increasing areal radius. Because of ourchoice of orientation we have e = ⋆u (the volume formon B is simply µB = u ∧ e). Using this expression fore and the definition of the Kodama vector field (D20)we have g(k,u) = 〈k,u〉 = 〈⋆dR,u〉 = −〈dR, ⋆u〉 =−〈dR, e〉 = −dR(e) and hence, the Hodge star of theKodama current becomes

⋆ j = −pdR(u)u− dR(e)e , (D42)

which, inserted in (D23), gives the following expressionfor the differential of the MS energy for a perfect fluid:

dE = −4πR2(

pdR(u)u+ dR(e)e)

. (D43)

Hence, the variation of the MS energy along u and e is,respectively:

dE(u) = −4πR2 pdR(u) , (D44a)

dE(e) = +4πR2 dR(e) . (D44b)

These expressions have a good physical interpretation:Since the matter moves along u, (D44a) expresses thefact that the energy can only increase (decrease) if themotion along u does (releases) work against (with) theaction of the pressure. Equation (D44b) expresses thealmost obvious increase (decrease) of gravitational masswith increase (decrease) of volume in the rest system ofthe matter. We said ‘almost’ because 4πR2dR(e) is notquite the increment of proper volume. The difference ac-counts for the fact that kinetic and gravitational bindingenergy are themselves gravitationally active. To see thatthis is indeed what (D44b) implies, let p be some point inspacetime and Sp the two-sphere of spherical symmetrythrough p. Assume Sp to have a regular interior, thatis, that Sp bounds a 3-ball Bp in the hypersurface Σ or-thogonal to u. Except for the origin of Bp, we can writeBp = σ×S2, where σ is a spacelike curve in B orthogonalto u, going from the center of symmetry to π(p). Usingthe expression E = (R/2)(1 + (dR(u))2 − (dR(e))2) forthe MS energy to eliminate dR(e) in (D44b), integrating

37

the latter over σ, and re-expressing the result as a volumeintegral, one gets:

E(p) =

∫

Bp

(

1 + (dR(u))2 − 2E

R

)1/2

µΣ . (D45)

One sees that the MS energy contains the contributionfrom the proper mass contained in the ball Bp,

M(p) =

∫

Bp

µΣ , (D46)

as well as contributions from the ‘kinetic’ and ‘poten-tial’ energy (Hayward, 1996; Misner and Sharp, 1964).In a Newtonian approximation, that is for small ‘veloc-ity’ dR(u) and weak field (small E/R) one can expandthe square root in (D45) and gets, in leading order:

E(p) ≈∫

Bp

(

+ 12(dR(u))2 − M

R

)

µΣ . (D47)

In this approximation the MS energy is therefore justthe sum of the proper mass and the Newtonian kineticand potential energies contained in the ball Bp. Thisprovides a sound justification for the interpretation ofthe MS energy as the active gravitational energy.At this point we can compute also the differentials of

the two parts (D34) of the MS energy separately. Thedifferential of the Ricci part follows directly from (D40):

dER = 4πR2(dR+ 1

3Rd)

(D48)

and the differential of the Weyl part is just the differenceof this with (D43):

dEW = −4πR2(

(+ p)dR(u)u+ 13Rd

)

. (D49)

Its components in the directions u and e are then

dEW(u) = −4πR2(+ p)dR(u)− 4π3 R3d(u) , (D50a)

dEW(e) = − 4π3 R3d(e) . (D50b)

It is now instructive to express the variation along uof the Ricci and Weyl parts of the MS energy in terms ofthe kinematical properties of the fluid velocity u. Recallthat, because of spherical symmetry, the rotation tensorvanishes identically and the shear tensor has only oneindependent component. The kinematical quantities re-duces thus to two scalars: the expansion

θ := divu (D51)

and the shear scalar

σ :=dR(u)

R− 1

3θ . (D52)

The shear tensor is then given by the trace-free endomor-phism σ = σ(QS2−2Qe), whereQS2 andQe denote, re-spectively, the projections onto the two-dimensional sub-space of T (M) tangential to the two-sphere and onto

the one-dimensional space parallel to e (for the lattersee (C1a)). We recall that the divergence-freeness of theenergy-momentum tensor (D38) is equivalent to

(+ p) θ = −d(u) (D53a)

(+ p) b = −dp(e) , (D53b)

where b := −g(∇uu, e) is the acceleration (scalar) of uin radial direction (the minus sign in the latter formulais because the metric is negative definite in spatial direc-tions).Now, using (D53a) and (D52) we get:

dER(u) =4π

3R3(3 σ − p θ) (D54a)

dEW(u) = −4π

3R3(+ p)3σ (D54b)

With the equations just derived we can now say whenthe MS energy, and its Ricci and Weyl parts, are tem-porally or spatially constant. Here, by temporally (spa-tially) constant we mean that the variation in directionof u (e) vanishes. We collect the results in the following

Theorem 4. Consider a spherically symmetric fluid with+p 6= 0 and restrict to the region where dR is spacelike.Then for the MS energy E and its Ricci and Weyl partsER and EW the following statements hold true:(i) E is temporally constant iff p = 0 or dR(u) = 0;(ii) E is spatially constant iff = 0;(iii) EW is temporally constant iff σ = 0;(iv) EW is spatially constant iff is spatially constant;(v) ER is temporally (spatially) constant iff R3 is tem-

porally (spatially) constant.

The proof is a straightforward application of the formulaejust derived above. Note that the assumption + p 6=0 is needed only for (iii). The assumption that dR isspacelike is needed only for (ii): If dR is spacelike, thenfor any spacelike spherically symmetric vector e (hencetangent to the basis manifold B) it holds dR(e) 6= 0,since in a two-dimensional Lorentzian manifold any twospacelike vectors are linearly dependent.

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